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ALGEBRAIC CALDERON-ZYGMUND THEORY

MARIUS JUNGE, TAO MEI,JAVIER PARCET AND RUNLIAN XIA

Abstract. Calderon-Zygmund theory has been traditionally developed on

metric measure spaces satisfying additional regularity properties. In the lackof good metrics, we introduce a new approach for general measure spaces whichadmit a Markov semigroup satisfying purely algebraic assumptions. We shall

construct an abstract form of ‘Markov metric’ governing the Markov processand the naturally associated BMO spaces, which interpolate with the Lp-scaleand admit endpoint inequalities for Calderon-Zygmund operators. Motivated

by noncommutative harmonic analysis, this approach gives the first form ofCalderon-Zygmund theory for arbitrary von Neumann algebras, but is alsovalid in classical settings like Riemannian manifolds with nonnegative Ricci

curvature or doubling/nondoubling spaces. Other less standard commutativescenarios like fractals or abstract probability spaces are also included. Among

our applications in the noncommutative setting, we improve recent results

for quantum Euclidean spaces and group von Neumann algebras, respectivelylinked to noncommutative geometry and geometric group theory.

Introduction

The analysis of linear operators associated to singular kernels is a central topicin harmonic analysis and partial differential equations. A large subfamily of thesemaps is under the scope of Calderon-Zygmund theory, which exploits the relationbetween metric and measure in the underlying space to provide sufficient conditionsfor Lp boundedness. The Calderon-Zygmund decomposition [6] or the Hormandersmoothness condition for the kernel [29] combine the notions of proximity in termsof the metric with that of smallness in terms of the measure. The doubling and/orpolynomial growth conditions between metric and measure yield more general formsof the theory [14, 46, 47, 62, 63]. To the best of our knowledge, the existence of ametric in the underlying space is always assumed in the literature.

In this paper, we introduce a new form of Calderon-Zygmund theory for generalmeasure spaces admitting a Markov semigroup which only satisfies purely algebraicassumptions. This is especially interesting for measure spaces where the geometricinformation is poor. It includes abstract probability spaces or fractals like theSierpinski gasket, where a Dirichlet form is defined. It is also worth mentioning thatour approach recovers Calderon-Zygmund theory for classical spaces and providesalternative forms over them. In spite of these promising directions —very littleexplored here— our main motivation has been to develop a noncommutative formof Calderon-Zygmund theory for noncommutative measure spaces (von Neumannalgebras) where the notions of quantum metric [37, 55, 56] seem inefficient.

1

2 JUNGE, MEI, PARCET AND XIA

A great effort has been done over the last years to produce partial results towardsa noncommutative Calderon-Zygmund theory [24, 28, 33, 45, 49]. The model casesconsidered so far are all limited to (different) noncommutative forms of Euclideanspaces, described as follows:

A) Tensor products. Let f = (fjk) : Rn → B(ℓ2) be a matrix-valued function andconsider the tensor product extension of a standard Calderon-Zygmund operatoracting on f , formally given by

Tf(x) =

∫Rn

k(x, y)f(y) dy =(Tfjk(x)

)jk

for x /∈ suppf.

The Lp-boundedness of this map in the associated (tensor product) von Neumannalgebra M = L∞(Rn)⊗B(ℓ2) trivially follows for p > 1 from the vector-valuedtheory, due to the UMD nature of Schatten p-classes. On the contrary, no endpointestimate for p = 1 is possible using vector-valued methods. The original argumentin [49] —also in a recent simpler form [4]— combines noncommutative martingaleswith a pseudolocalization principle for classical Calderon-Zygmund operators. Moreprecisely, a quantification of how much L2-mass of a singular integral is concentratedaround the support of the function on which it acts. This inequality has been thekey tool in the recent solution of the Nazarov-Peller conjecture [8], a strengtheningof the celebrated Krein conjecture [59] on operator Lipschitz functions.

B) Crossed products. New Lp estimates for Fourier multipliers in group vonNeumann algebras have recently gained considerable momentum for its connectionsto geometric group theory. The first Hormander-Mikhlin type theorem in thisdirection [33] exploited finite-dimensional cocycles of the given group G to transferthe problem to the cocycle Hilbert space H = Rn. To find sufficient regularityfor Lp-boundedness amounts to study Calderon-Zygmund operators in the crossedproducts L∞(Rn)⋊G induced by the cocycle action. Nonequivariant extensions ofCZOs on these von Neumann algebras were investigated in [33], after identifyingthe right BMO space for the length function determined by the cocycle. Theseoperators have the form∫

G

fg ⋊ λ(g) dµ(g) 7→∫G

Tg(fg)⋊ λ(g) dµ(g).

Here µ, λ respectively denote the Haar measure and left regular representation onthe (unimodular) group G, whereas Tg = αgTαg−1 is a twisted form of a classicalCZO T on Rn by the cocycle action α. We refer to [34, 50] for further results.

C) Quantum deformations. PDEs in matrix algebras and ‘noncommutativemanifolds’ appear naturally in theoretical physics. Pseudodifferential operatorswere introduced by Connes in 1980 to study a quantum form of the Atiyah-Singerindex theorem over these algebras. These techniques have been underexploited overthe last 30 years, due to fundamental obstructions to understand singular integraltheory in this context. The core of singular integrals and pseudodifferential operatorLp-theory was developed in [24] over the archetypal algebras of noncommutativegeometry. It includes quantum tori, Heisenberg-Weyl algebras and other quantumdeformations of Rn of great interest in quantum field theory, string theory andquantum probability. This was the first approach to a ‘fully noncommutative’Calderon-Zygmund theory for CZOs not acting on copies of Rn as tensor or crossedproduct factors, but still related to Euclidean methods.

ALGEBRAIC CALDERON-ZYGMUND THEORY 3

We introduce in this paper the first form of Calderon-Zygmund theory validfor general (semifinite) von Neumann algebras. As we explained above, the maindifficulty arises from the lack of very standard geometric tools, like the existenceof a nice underlying metric or the construction of suitable covering lemmas. Weshall circumvent it using a very different approach based on algebraic properties ofa given Markov process. Our applications cover a wide variety of scenarios whichwill be discussed, giving especial emphasis to noncommutative forms of Euclideanspaces and locally compact abelian groups, which are our main classical models. Inthe first case, we shall weaken/optimize the CZ conditions on quantum Euclideanspaces [24]. In the second case, LCA groups correspond to quantum groups whichare both commutative and cocommutative [60]. We shall give CZ conditions forconvolution maps over quantum groups. In the cocommutative (non necessarilycommutative) context, this includes group von Neumann algebras.

Calderon-Zygmund extrapolation

Based on the behavior of the Hilbert transform in the real line, the main goalof Calderon-Zygmund theory is to establish regularity properties on the kernel ofa singular integral operator, so that L2-boundedness automatically extrapolates toLp boundedness for 1 < p < ∞. A singular integral operator in a Riemannianmanifold (X,d, µ) admits the kernel representation

Tkf(x) =

∫X

k(x, y)f(y) dµ(y) for x /∈ suppf.

Namely, Tk is only assumed a priori to send test functions into distributions, so thatit admits a distributional kernel in X × X which coincides in turn with a locallyintegrable function k away from the diagonal x = y, where the kernel presentscertain singularity. This already justifies the assumption x /∈ suppf in the kernelrepresentation. The work in [6, 29] culminated in the following sufficient conditionson a singular integral operator in Rn for its Lp-boundedness:

i) L2-boundedness ∥∥Tk : L2(Rn) → L2(Rn)∥∥ <∞.

ii) Hormander kernel smoothness

ess supx,y∈Rn

∫|x−z|>2|x−y|

∣∣k(x, z)− k(y, z)∣∣+ ∣∣k(z, x)− k(z, y)

∣∣ dz <∞.

Historically, this was used to prove a weak endpoint inequality in L1. The sameholds for Riemannian manifolds with nonnegative Ricci curvature [1]. Alternativelyit is simpler to use L2-boundedness and the kernel smoothness condition to proveL∞ → BMO boundedness. The result then follows by well-known duality andinterpolation arguments. Our strategy resembles this approach:

P1. Identify the appropriate BMO spaces.P2. Prove the expected interpolation results with Lp spaces.P3. Provide conditions on CZO’s which yield L∞ → BMO boundedness.

In the classical setting, we typically find H1/BMO spaces associated to a metric or amartingale filtration. Duong and Yan [19, 20] extended this theory replacing someaverages over balls in the metric space by semigroups of positive operators, although


the existence of a metric was still assumed. This assumption was removed in [32, 44]providing a theory of semigroup type BMO spaces with no further assumptions onthe given space. In particular, we could say that Problems 1 and 2 were solved in[32], but it has been unclear since then how to provide natural CZ conditions whichimply L∞ → BMO estimates. In this paper we solve P3 by splitting it into:

P3a. Construct a ‘metric’ governing the Markov process.P3b. Define ‘metric BMO’ spaces which still interpolate with the Lp scale.P3c. Provide CZ conditions giving L∞ → BMO boundedness for metric BMO’s.

P3a. Markov metrics. Given a Markov semigroup S = (St)t≥0 on the semifinitevon Neumann algebra (M, τ) —in other words, formed by normal self-adjoint cpumaps St— we introduce a Markov metric for it as any family

Q =(Rj,t, σj,t, γj,t

): (j, t) ∈ Z+ × R+

composed of completely positive unital (cpu) maps Rj,t : M → M and elementsσj,t, γj,t of M with γj,t ≥ 1M, such that the following estimates (which show howthe Markov metric governs the Markov semigroup in a controlled way) hold:

i) Hilbert module majorization:⟨ξ, ξ⟩St

≤∑j≥1

σ∗j,t

⟨ξ, ξ⟩Rj,t

σj,t.

ii) Metric integrability condition: kQ = supt>0

∥∥∥∑j≥1

σ∗j,tγ

2j,tσj,t

∥∥∥ 12

M< ∞.

Here ⟨ , ⟩Φ is the M-valued inner product on M⊗M for any cpu map Φ, given by⟨a ⊗ b, a′ ⊗ b′⟩Φ = b∗Φ(a∗a′)b′. Markov metrics are a priori unrelated to Rieffel’squantum metric spaces [55, 56]. They present on the contrary vague similaritieswith abstract formulations of classical CZ theory in the absence of CZ kernelsand/or doubling measures [2, 62]. We shall explain what motivates our definitionbelow and we shall also illustrate how Euclidean and other classical metrics fit in.

P3b. Metric type BMO spaces. Let

∥f∥BMOcS= sup

t≥0

∥∥∥(St(f∗f)− (Stf)

∗(Stf)) 1

2∥∥∥M

and ∥f∥BMOS = max∥f∥BMOcS, ∥f∗∥BMOc

S. We shall define the semigroup type

BMO space BMOS(M) as the weak-∗ closure of M in certain direct sum of Hilbertmodules determined by S = (St)t≥0. These spaces interpolate with the Lp scale[32]. Given a Markov metricQ associated to this semigroup, let us define in addition

∥f∥BMOQ = max∥f∥BMOc

Q, ∥f∗∥BMOc

Q

,

∥f∥BMOcQ= sup

t>0inf

Mt cpuMt:M→M

supj≥1

∥∥∥(γ−1j,t

[Rj,t|f |2−|Rj,tf |2+ |Rj,tf−Mtf |2

]γ−1j,t

) 12∥∥∥M.

Theorem A1. Let (M, τ) be a semifinite von Neumann algebra equipped with aMarkov semigroup S = (St)t≥0. Let us consider a Markov metric Q associated toS = (St)t≥0. Then, we find

∥f∥BMOS ≲ kQ ∥f∥BMOQ .

Thus, defining BMOQ(M) as a subspace of BMOS(M), it interpolates with Lp(M).


Theorem A1 solves P3b. Its proof is not hard after having defined the rightnotion of Markov metric and the right BMO norm. Let us note in passing that theterm Rj,tf −Mtf is there to accommodate nondoubling spaces to our definition inthe spirit of Tolsa’s RBMO space [62]. As a consequence of Theorem A1, provingL∞ → BMO boundedness for metric BMO’s (Problem 3c) implies the same resultfor semigroup BMO spaces (Problem 3). Of course, one could try to prove such astatement directly, but it seems that the metric/measure relation found with thesenew notions is crucial for a noncommutative CZ theory.

P3c. Calderon-Zygmund operators. The commutative idea behind the notionof Markov metric (explained in more detail in the body of the paper) is to findpointwise majorants of the integral kernels of our semigroup S = (St)t≥0, so thatwe can dominate St by certain sum of averaging operators over a distinguishedfamily of measurable sets Σj,t(x). These sets may be considered as the ‘balls’ inthe Markov metric. In the noncommutative setting, this pointwise estimates mustbe written in terms of the given Hilbert module majorization and the cpu maps Rj,t

must be averages over certain projections qj,t. Making this precise in full generalityis one of the challenges of our algebraic approach and too technical to be explainedat this point of the paper. A simple model case is given by

(Avg) Rj,tf = (id⊗ τ)(qj,t)− 1

2 (id⊗ τ)(qj,t(1⊗ f)qj,t

)(id⊗ τ)(qj,t)

− 12

for certain family of projections qj,t ∈ M⊗M. The linear map Rj,t(1⊗ f) = Rj,tftrivially amplifies to M⊗M. We may also consider similar formulas for the cpumaps Mt in the metric BMO norm. (Avg) allows to identify the Markov metric interms of the ‘balls’ qj,t instead of the corresponding averaging maps Rj,t.

As it happens in classical Calderon-Zygmund theory, we need to impose someadditional properties in the Markov metric to establish a good relation with theunderlying (noncommutative) measure. We have split these into algebraic andanalytic conditions, further details will be given in the text. Let us just mentionthat the algebraic ones are inherent to noncommutativity and hold trivially incommutative cases. The analytic ones provide forms of Jensen’s inequality anda measure/metric growth condition. Once we know the Markov metric satisfiesthese conditions, we may introduce Calderon-Zygmund operators. Assume thatT (AM) ⊂ M for a map T acting on a weak-∗ dense subalgebra AM ⊂ M. Thegoal is to establish sufficient Calderon-Zygmund conditions on T for L∞ → BMOc

boundedness. These are noncommutative forms of standard properties. Again, itis unnecessary to introduce them here in full generality, we do it in Section 2. Inthe model case above, our CZ conditions are:

i) L∞(Lc2)-boundedness∥∥∥(id⊗ τ)

((id⊗ T )(x)∗(id⊗ T )(x)

) 12∥∥∥M

≲∥∥(id⊗ τ)(x∗x)

12

∥∥M.

ii) Size ‘kernel’ conditions

• Mt

(∣∣(id⊗ T )((1⊗ f)(Aj,t − at)

)∣∣2) ≲ γ2j,t∥f∥2M,

• Rj,t

(∣∣(id⊗ T )((1⊗ f)(Aj,t − aj,t)

)∣∣2) ≲ γ2j,t∥f∥2M,

for certain family of operators Aj,t, aj,t ∈ M⊗M with Aj,t ≥ aj,t.


iii) Hormander ‘kernel’ conditions

• Φj,t

(∣∣δ((id⊗ T )((1⊗ f)(1 − aj,t)

))∣∣2) ≲ γ2j,t∥f∥2M,

• Ψj,t

(∣∣δ((id⊗ T )((1⊗ f)(1−Aj,t)

))∣∣2) ≲ γ2j,t∥f∥2M,

for certain family of cpu linear maps Φj,t,Ψj,t : M⊗M → M.

In condition ii), Aj,t and aj,t play the role of ‘dilated balls’ from qj,t. In the lastcondition, δ is the derivation x 7→ x ⊗ 1 − 1 ⊗ x acting on the second leg of thetensor product. In the Euclidean case, these conditions reduce to L2-boundednessand the classical size/smoothness conditions for the kernel. Our general conditionsinclude many more amplification algebras and derivations, other than M⊗M andδ. Any map T : AM → M satisfying the above CZ-conditions will be called acolumn CZ-operator.

Theorem A2 . Let (M, τ) be a semifinite von Neumann algebra equipped witha Markov semigroup S = (St)t≥0 with associated Markov metric Q fulfilling ouralgebraic and analytic assumptions. Then, any column CZ-operator T defines abounded operator

T : AM → BMOcQ(M).

Interpolation and duality give similar (symmetrized) conditions for Lp-boundedness.

A generalized form of Theorem A2 is the main result of this paper. It is easyto recover Euclidean CZ-extrapolation from it. In the Euclidean and many otherdoubling scenarios, the size kernel condition ii) does not play any role. Our nextgoal is to explore how the general form of Theorem A2 applies in concrete vonNeumann algebras with specific Markov metrics.

Applications

Algebraic Calderon-Zygmund theory applies in classical and noncommutativemeasure spaces. In the commutative context, we shall limit ourselves to provethat algebraic and classical theories match in three important cases: Euclideanspaces with both Lebesgue or Gaussian measures and Riemannian manifolds withnon-negative Ricci curvature. We shall not explore further implications in newcommutative scenarios, like abstract probability spaces or fractals equipped withspecific Dirichlet forms. In the noncommutative context, we start by analyzingthe model case of matrix-valued functions from a very general viewpoint. We alsoconsider Calderon-Zygmund operators over matrix algebras, generalizing triangulartruncations as the archetype of singular integral operator. Most importantly, ourabstract theory applies to quantum Euclidean spaces and quantum groups, whichconstitute our main motivations in this paper.

It will be useful to specify the form that our Calderon-Zygmund operators takewhen come associated to a concrete kernel. Our applications below include CZconditions on the kernel. In the basic model case above, we set

(Ker 1) Tkf = (id⊗ τ)(k(1⊗ f)

)for some kernel k affiliated to M⊗Mop. Recall that the opposite structure (Mop

is the same algebra M endowed with the reversed product a · b = ba) in the second


tensor leg of the kernel for this (standard) model was already justified in [24]. It isa feature of CZ theory which can only be witnessed in noncommutative algebras. Itwill also be useful to generalize a bit our model case before analyzing any concreteapplication. Consider an auxiliary von Neumann algebra A equipped with a n.s.f.trace φ, a ∗-homomorphism σ : M → A⊗M and the representation

(Ker 2) Skf = (id⊗ φ)(k flip σ(f)

)for some kernel k affiliated to M⊗Aop. Of course, when A = M and σ(f) = 1⊗ fwe recover our model case above, with kernel representation (Ker 1). This moregeneral framework requires to redefine Rj,t in (Avg) and the CZ conditions, as weshall do in the body of the paper. The advantage is to take A as an elementary(commutative) algebra, from which we can transfer metric information. One maythink of σ as a corepresentation in the terminology of quantum groups. TheoremA2 still holds in this case. We shall refer to intrinsic or transferred theories whenusing the model case A = M or its generalization respectively.

Quantum Euclidean spaces. As geometrical spaces with noncommuting spatialcoordinates, quantum Euclidean spaces have appeared frequently in the literatureof mathematical physics, in the contexts of string theory and noncommutativefield theory. These algebras play the role of a central and testing example innoncommutative geometry as well. The singular integral operators on quantumEuclidean spaces naturally appear in the recent study of Connes’ quantized calculus[40, 42, 58] and noncommutative harmonic analysis [11, 24, 25, 65]. Let

Θ ∈Mn(R)

be anti-symmetric. Briefly, the quantum Euclidean space RΘ is the von Neumannalgebra generated by certain family of unitaries uj(s) : 1 ≤ j ≤ n, s ∈ R satisfying

uj(s)uj(t) = uj(s+ t),

uj(s)uk(t) = e2πiΘjkstuk(t)uj(s).

Define λΘ(ξ) = u1(ξ1)u2(ξ2) · · ·un(ξn) and set

f =

∫Rn

fΘ(ξ)λΘ(ξ) dξ = λΘ(fΘ).

for fΘ ∈ Cc(Rn). The trace on RΘ is determined by

τΘ(f) = τΘ

(∫Rn

fΘ(ξ)λΘ(ξ) dξ

)= fΘ(0).

When Θ = 0, Lp(RΘ, τΘ) reduces to Lp(Rn) with the Lebesgue measure. Precisedefinitions and a theory of singular integrals forRΘ appears in [24]. The main resultrelies on gradient kernel conditions for the intrinsic model (Ker 1). Remarkably, weshow in this paper that the transference model (Ker 2)

σΘ : RΘ ∋ λΘ(ξ) 7→ expξ ⊗λΘ(ξ) ∈ L∞(Rn)⊗RΘ

goes further, since it just requires Hormander type smoothness for the kernel. Hereexpξ stands for the ξ-th character exp(2πi⟨ξ, ·⟩) in Rn. There is a close relationbetween both models in this case

Tk(f) = Sk(f) for k = πΘ(k) and πΘ(m⊗ expξ) = mλΘ(ξ)∗ ⊗ λΘ(ξ).


Another crucial map is the ∗-homomorphism

πΘ : L∞(Rn) ∋ expξ 7→ λΘ(ξ)⊗ λΘ(ξ)∗ ∈ RΘ⊗Rop

Θ .

If BR denotes the Euclidean R-ball centered at the origin, define the projectionsaR = πΘ(15BR

) and a⊥R = 1− aR. Set kσ = (σΘ ⊗ idRopΘ)(k) ∈ L∞(Rn)⊗RΘ⊗Rop

Θ

and define the derivation δφ(x, y) = φ(x) − φ(y) to set the kernel condition inL∞(Rn)⊗RΘ⊗Rop

Θ

(Hor) sup|x|≤R,|y|≤R

∣∣∣δ((id⊗ id⊗ τΘ)[kσ(1⊗ 1⊗ f)(1⊗ a⊥R)

])(x, y)

∣∣∣ ≲ ∥f∥RΘ.

As we shall justify in the paper, (Hor) is the right form of Hormander kernelcondition in this framework. The column BMO-norm admits in RΘ an equivalentform

∥f∥BMOc(RΘ) ≈ ∥σΘ(f)∥BMOc(Rn;RΘ)

for the operator-valued BMO space BMOc(Rn;RΘ) from [43]. These are all theingredients to obtain Calderon-Zygmund extrapolation over quantum Euclideanspaces. Namely, the general form of Theorem A2 then yields the following theorem.

Theorem B1. Tk is bounded from RΘ to BMOc(RΘ) provided :

i) Tk is bounded on L2(RΘ).ii) The kernel condition (Hor) holds.

Interpolation and duality give similar (symmetrized) conditions for Lp-boundedness.

Theorem B1 improves the main CZ extrapolation theorem in [24] by reducing thegradient kernel condition there to the (more flexible) Hormander integral conditionabove, as we shall prove along the paper. In fact, the result which we shall finallyprove is slightly more general than the statement above.

Quantum groups. Let G be a locally compact group with a left invariant Haarmeasure µ. When G is abelian, the Fourier transform carries the convolution algebra

L1(G, µ) into the multiplication algebra L∞(G, µ) associated to the dual groupwith its (normalized) Haar measure. However, when G is not abelian, we can notconstruct the dual group and the multiplication algebra above becomes the groupvon Neumann algebra which is generated by the left regular representation of G.These algebras are basic models of (noncommutative, but still cocommutative)quantum groups, over which we shall study singular integrals.

Let G be a locally compact quantum group —precise definitions in the bodyof the paper— with comultiplication ∆ and left-invariant and right-invariant Haarweights ψ, φ. Given a weak-∗ dense subspace A of L∞(G) and a linear map Tsatisfying T (A) ⊂ L∞(G), it is is a convolution map when

(T ⊗ idG) ∆ = ∆ T = (idG ⊗ T ) ∆.To simplify the problem, we shall consider the case where G admits an α-doublingintrinsic Markov metric. That is, the projections which generate the cpu mapsRj,t’s satisfy

ψ(qα(j),t)

ψ(qj,t)≤ cα

for a strictly increasing function α : N → N with α(j) > j and a constant cα.


Theorem B2. Let G be a locally compact quantum group and assume it comesequipped with a convolution semigroup S = (St)t≥0 which admits an α-doublingintrinsic Markov metric. Let T : A → L∞(G) be a convolution map defined on aweakly dense ∗-subalgebra A of L∞(G) such that

i) T is bounded on L2(G).

ii)1

|ψ(qj,t)|2(ψ ⊗ ψ)

((qj,t ⊗ qj,t)

∣∣δ(T (fq⊥α(j),t)∣∣2) ≲ ∥f∥2L∞(G).

Then, the linear map T extends to a bounded map T : L∞(G) → BMOcS(L∞(G)).

As usual, Lp estimates follow from symmetrized conditions by interpolation andduality. In fact, we shall prove a more general statement which incorporates tensorproducts with an additional algebra (M, τ). Theorem B2 is proved one more timefrom Theorem A2. In fact, it is conceivable to remove the α-doubling restriction andstill make the convolution map bounded under an additional size kernel conditionas Theorem A2 indicates.

Noncommutative transference. In a different direction, we shall finish thispaper with a section devoted to noncommutative forms of Calderon-Cotlar methodof transference [5, 13, 15]. The basic idea is to transfer Lp estimates of convolutionmaps on quantum groups to a much wider class of maps which arise by transference.We refer to [7, 9, 11, 48, 50, 54] for other forms of transference in the context ofgroup von Neumann algebras and quantum tori.

1. Markov metrics

An abstract form of Calderon-Zygmund theory incorporating noncommutativealgebras lacks standard geometrical tools. Given a Markov semigroup on a vonNeumann algebra —a semigroup of normal cpu self-adjoint maps on the givenalgebra— we shall construct a ‘metric’ governing the Markov process. Our modelcase in a commutative measure space (Ω, µ) is a Markov semigroup of linear mapsof the form

Stf(x) =

∫Ω

st(x, y)f(y) dµ(y).

The idea is to find pointwise majorants of the form

(1.1) st(x, y) ≤∞∑j=1

|σj,t(x)|2

µ(Σj,t(x))χΣj,t(x)(y),

so that Stf(x) is dominated by a given combination of averaging operators overcertain measurable sets Σj,t(x). These sets will determine some sort of metric on(Ω, µ) under additional integrability properties. Naively, we may think of them asballs or coronas around x in the hidden metric with radii depending on (j, t). Inthis section we formalize this idea and construct BMO spaces with respect to theassociated ‘Markov metric’ which satisfy the expected interpolation results.

1.1. Hilbert modules. A noncommutative measure space is a pair (M, τ) formedby a semifinite von Neumann algebra M and a n.s.f. trace τ . We assume in whatfollows that the reader is familiar with basic terminology from noncommutativeintegration theory [36, 61]. Nonexpert readers may proceed by fixing a measure


space (Ω, µ) with M = L∞(Ω) and τ the integral operator associated to µ. Givena cpu map Φ : M → M we may construct the Hilbert module M⊗ΦM. Namelyconsider the seminorm on M⊗M

∥ξ∥M⊗ΦM =∥∥√⟨ξ, ξ⟩Φ

∥∥M

determined by the M-valued inner product⟨∑jaj ⊗ bj ,

∑ka′k ⊗ b′k

⟩Φ=∑

j,kb∗jΦ(a

∗ja

′k)b

′k.

Then M⊗ΦM will stand for the completion in the topology determined by ξα → ξwhen τ(⟨ξ− ξα, ξ− ξα⟩Φ g) → 0 for all g ∈ L1(M). When Φ is normal, the abstractcharacterization of Hilbert modules [51] yields a weak-∗ continuous rightM-modulemap ρ : M⊗ΦM → Hc⊗M satisfying ⟨ξ, η⟩Φ = ρ(ξ)∗ρ(η). Let us collect a fewproperties which will be instrumental along this paper.

Lemma 1.1. Given a cpu map Φ : M → M :

i)⟨ξ1 + ξ2, ξ1 + ξ2

⟩Φ≤ 2⟨ξ1, ξ1

⟩Φ+ 2⟨ξ2, ξ2

⟩Φ,

ii)∥∥f ⊗ 1M − 1M ⊗ Φf

∥∥M⊗ΦM =

∥∥Φ|f |2 − |Φf |2∥∥ 1

2

M,

iii)∣∣Φf − g

∣∣2 ≤⟨f ⊗ 1M − 1M ⊗ g, f ⊗ 1M − 1M ⊗ g

⟩Φ,

iv)∥∥f ⊗ 1M − 1M ⊗ Φf

∥∥M⊗ΦM ∼ inf

g∈M

∥∥f ⊗ 1M − 1M ⊗ g∥∥M⊗ΦM,

v) If Φ ≤cp

∑k βkΨk, then

∥ξ∥M⊗ΦM ≤(∑

kβk∥ξ∥2M⊗Ψk

M

) 12

.

Proof. The first inequality follows from hermitianity of the inner product and theidentity ⟨ξ, η⟩Φ = ρ(ξ)∗ρ(η) explained above. The second one is straightforwardfrom the definition of M⊗ΦM. The third inequality follows from Kadison-Schwarzinequality after expanding both sides. The lower estimate in iv) holds trivially withconstant 1, while the upper estimate holds with constant 2 since

f ⊗ 1M − 1M ⊗ Φf = (f ⊗ 1M − 1M ⊗ g)− (1M ⊗ (Φf − g))

and the second term on the right hand side is estimated using iii). Finally, for thelast inequality let ξ =

∑k ak⊗Bj and define the column matrices A∗ =

∑k a

∗k⊗ek1

and B =∑

k Bj ⊗ ek1. Then we find⟨ξ, ξ⟩Φ

=∑

j,kb∗jΦ(a

∗jak)Bj = B∗Φ(A∗A)B

≤∑

kβkB

∗Ψk(A∗A)B =

∑kβk⟨ξ, ξ⟩Ψk.

Let (M, τ) denote a noncommutative measure space equipped with a Markovsemigroup S = (St)t≥0 acting on it. A Markov metric associated to (M, τ) and Sis determined by a family

Q =(Rj,t, σj,t, γj,t

): (j, t) ∈ Z+ × R+

where Rj,t : M → M are completely positive unital maps and σj,t, γj,t are elementsof the von Neumann algebra M with γj,t ≥ 1M, so that the estimates below hold:

i) Hilbert module majorization:⟨ξ, ξ⟩St

≤∑j≥1

σ∗j,t

⟨ξ, ξ⟩Rj,t

σj,t,


ii) Metric integrability condition: kQ = supt>0

∥∥∥∑j≥1

σ∗j,tγ

2j,tσj,t

∥∥∥ 12

M< ∞.

Our notion of Markov metric is easily understood for our (commutative) modelcase above. Let S = (St)t≥0 be a Markov semigroup on (Ω, µ) with associatedkernels st(x, y) satisfying the pointwise estimate (1.1). Given ξ : Ω × Ω → Cessentially bounded, we have

(1.2)⟨ξ, ξ⟩St=

∫Ω

st(x, y)|ξ(x, y)|2 dµ(y) ≤∞∑j=1

|σj,t(x)|2

µ(Σj,t(x))

∫Σj,t(x)

|ξ(x, y)|2 dµ(y).

This means that Rj,tf(x) is the average of f over the set Σj,t(x). Reciprocally, if wetake ξk(x, y) = ϕk(y0−y) to be an approximation of identity around y0, we recoverthe pointwise estimates for the kernel st(x, y0). In other more general contexts, theupper bounds for the kernel or even the kernel description of the semigroup mightnot have the same form. As we shall see, many of these cases can still be handledvia Hilbert module majorization. We shall provide along the paper a wide varietyof examples which fall into these possible classes.

1.2. Semigroup BMOs. Given a noncommutative measure space (M, τ) and aMarkov semigroup S = (St)t≥0 acting on (M, τ), we may define the semigroupBMOS -norm as

∥f∥BMOS = max∥f∥BMOr

S, ∥f∥BMOc

S

,

where the row and column BMO norms are given by

∥f∥BMOrS

= supt≥0

∥∥∥(St(ff∗)− (Stf)(Stf)

∗) 1

2∥∥∥M,

∥f∥BMOcS

= supt≥0

∥∥∥(St(f∗f)− (Stf)

∗(Stf)) 1

2∥∥∥M.

This definition makes sense since we know from the Kadison-Schwarz inequalitythat |Stf |2 ≤ St|f |2. The null space of this seminorm is kerA∞, the fixed-pointsubspace of our semigroup. Indeed, if ∥f∥BMOS = 0 we know from [12] that fbelongs to the multiplicative domain of St, so that

τ(gf) = τ(St/2(gf)) = τ(St/2(g)St/2(f)) = τ(gSt(f)).

This proves that f is fixed by the semigroup. Reciprocally, kerA∞ is a ∗-subalgebraof M by [35]. Thus, the seminorm vanishes on kerA∞. In particular, we obtain anorm after quotienting out kerA∞. Letting wt(f) = f ⊗ 1− 1⊗ Stf , this providesus with a map

f ∈ M w7−→(wt(f)

)t≥0

∈⊕t≥0

M⊗StM

which becomes isometric when we equip M with the norm in BMOcS . Define BMOc

Sas the weak-∗ closure of w(M) in the latter space. Similarly, we may define BMOSas the intersection BMOr

S ∩BMOcS , where the row BMO follows by taking adjoints

above. The natural operator space structure is given by

Mm(BMOS(M)) = BMOS(Mm(M)) with St = idMm⊗ St.

Remark 1.2. Incidentally, we note that BMOS is written as bmo(S) in [32].


It will be essential for us to provide interpolation results between semigroup typeBMO spaces and the corresponding noncommutative Lp spaces. It is a hard problemto identify the minimal regularity on the semigroup S = (St)t≥0 which suffices forthis purpose. The first substantial progress was announced in a preliminary versionof [31], where the gradient form 2Γ(f1, f2) = A(f∗1 )f2 + f∗1A(f2) − A(f∗1 f2) withA the infinitesimal generator of the semigroup, was a key tool in finding sufficientregularity conditions in terms of nice enough Markov dilations. However we knowafter [32] an even sharper condition. Consider the sets

ASf =Btf =

1

t

(St(f

2) + f2 − St(f)f − fSt(f)): t > 0

,

Γ1M =f ∈ Ms.a. : ASf is relatively compact in L1(M)

,

where Ms.a. denotes the self-adjoint part of M. The family ASf is called uniformlyintegrable in L1(M) if for all ε > 0 there exists δ > 0 such that ∥(Btf)q∥1 < εfor every projection q satisfying τ(q) < δ. It is well-known that ASf is relativelycompact in L1(M) if and only if it is bounded and uniformly integrable. Let usalso recall that Btf → 2Γ(f, f) as t→ 0. Define

Lp(M) =

f ∈ Lp(M) : lim

t→∞Stf = 0

.

As it was explained in [32], the space Lp(M) is complemented in Lp(M) and

[L1(M),BMOS ] form an interpolation couple. A Markov semigroup S = (St)t≥0

satisfying that Γ1M is weak-∗ dense in Ms.a. is called regular. All the semigroupsthat we handle in this paper are regular. The following result will be crucial inwhat follows, we refer the reader to [32] for a detailed proof.

Theorem 1.3. If S = (St)t≥0 is regular on (M, τ)[BMOS , L

p(M)

]p/q

≃cb Lq(M) for all 1 ≤ p < q <∞.

Note that interpolation against the full space Lp(M) is meaningless since BMOSdoes not distinguish the fixed-point space of the semigroup. Very roughly, weshall typically apply the above result to a CZO which is bounded on L2(M) andsends a weak-∗ dense subalgebra A of M to BMOS . Recalling the projection mapJp : Lp(M) → L

p(M) and letting T denote the CZO, we find by interpolation andthe weak-∗ density of A that

JpT : Lp(M) =[A, L2(M)

]2/p

→[BMOS , L

2(M)

]2/p

= Lp(M) ⊂ Lp(M).

To obtain Lp boundedness of T , it suffices to assume that T leaves the fixed-pointspace invariant and is bounded on it. It should be noticed though, that in manycases the Lp boundedness of the CZO follows automatically. For instance, in Rn

with the Lebesgue measure and the heat semigroup, it turns out that Lp = Lp. On

the other hand, the fixed-point space for the Poisson semigroup on the n-torus is justcomposed of constant functions and the corresponding projection can be estimatedapart regarded as a conditional expectation. Moreover, the same applies for Fouriermultipliers on arbitrary discrete groups. The Lp boundedness for 1 < p < 2 willfollow by taking adjoints under certain symmetry on the hypotheses.


1.3. Markov metric BMOs. Let us now introduce a Markov metric type BMOspace for von Neumann algebras and relate it with the semigroup type BMO spacesdefined above. Given a Markov semigroup S = (St)t≥0 acting on (M, τ), considera Markov metric Q = (Rj,t, σj,t, γj,t) : (j, t) ∈ Z+ × R+ as defined above anddefine ∥f∥BMOQ = max∥f∥BMOc

Q, ∥f∗∥BMOc

Q, where the column BMO-norm is

given by

supt>0

infMt cpu

supj≥1

∥∥∥(γ−1j,t

[Rj,t|f |2 − |Rj,tf |2 + |Rj,tf −Mtf |2

]γ−1j,t

) 12∥∥∥M

and the infimum runs over cpu maps Mt : M → M. Since γj,t ≥ 1M, the inversesexist and L∞(M) embeds in BMOQ. Indeed, using that Rj,t and Mt are cpu, the

square braket above is bounded by 5∥f∥2∞1M and γ−2j,t ≤ 1M. The row norm is

estimated in the same way. Now, recalling the value of the constant kQ in ourdefinition of Markov metric, we prove that BMOQ embeds in BMOS .

Theorem 1.4. Let (M, τ) be a noncommutative measure space equipped with aMarkov semigroup S = (St)t≥0. Let us consider a Markov metric Q associated toS = (St)t≥0. Then, we find

∥f∥BMOS ≲ kQ ∥f∥BMOQ .

In particular, we see that L∞(M) ⊂ BMOQ ⊂ BMOS and[BMOQ, L

p(M)

]p/q

≃ Lq(M) for all 1 ≤ p < q <∞

for any Markov metric Q associated to a regular semigroup S = (St)t≥0 on (M, τ).

Proof. Let us set

ξt = f ⊗ 1M − 1M ⊗Mtf

= (f ⊗ 1M − 1M ⊗Rj,tf) + (1M ⊗ (Rj,tf −Mtf)) = ξ1j,t + ξ2j,t.

The assertion follows from Lemma 1.1 and our definition of Markov metric

∥f∥BMOcS

= supt>0

∥∥∥(St|f |2 − |Stf |2) 1

2∥∥∥M

= supt>0

∥∥f ⊗ 1M − 1M ⊗ Stf∥∥M⊗St

M

≲ supt>0

∥∥f ⊗ 1M − 1M ⊗Mtf∥∥M⊗St

M = supt>0

∥⟨ξt, ξt⟩St∥

12

M

≲ supt>0

∥∥∥∑j≥1

σ∗j,t

[⟨ξ1j,t, ξ1j,t⟩Rj,t

+ ⟨ξ2j,t, ξ2j,t⟩Rj,t

]σj,t

∥∥∥ 12

M

≤ kQ∥f∥BMOcQ.

The identities are clear. The first inequality follow from Lemma 1.1 iv), the secondone from the Hilbert module majorization associated to the Markov metric andLemma 1.1 i). To justify the last inequality, note that the square bracket inside theterm on the left equals Rj,t|f |2−|Rj,tf |2+|Rj,tf−Mtf |2. Hence, left multiplication

by γj,tγ−1j,t and right multiplication by γ−1

j,t γj,t yields the given inequality with kQthe metric integrability constant. The interpolation result follows from Theorem1.3 and the embeddings L∞(M) ⊂ BMOQ ⊂ BMOS . The proof is complete.


Remark 1.5. Let ξ =∑

j Aj ⊗Bj , with

Aj =(ajαβ

)and Bj =

(bjαβ

)elements of Mm(M). If St = idMm

⊗ St, it turns out that⟨ξ, ξ⟩St

=

(m∑

α=1

⟨∑j,β

ajαβ ⊗ bjβγ1︸︷︷︸ηα,γ1

,∑j,β

ajαβ ⊗ bjβγ2︸︷︷︸ηα,γ2

⟩St

)γ1,γ2

∈Mm(M).

This can be used to provide an operator space structure on BMOQ. Namely, thecanonical choice for the matrix norms is Mm(BMOQ(M)) = BMOQ(Mm(M))

where the Markov metric on Mm(M)

Q =(idMm

⊗Rj,t,1Mm⊗ σj,t,1Mm

⊗ γj,t)

is associated to the extended semigroup (St)t≥0. Then, we trivially obtain thatkQ = kQ < ∞. However, according to the identity above for ⟨ξ, ξ⟩St

, the Hilbertmodule majorization takes the form(

m∑α=1

⟨ηα,γ1

, ηα,γ2

⟩St

)γ1,γ2

≤∑j≥1

(σ∗j,t

m∑α=1

⟨ηα,γ1

, ηα,γ2

⟩Rj,t

σj,t

)γ1,γ2

.

This gives a matrix-valued generalization of our Hilbert module majorization forS = (St)t≥0 on M, to be checked when we use this o.s.s. Theorem 1.4 yieldsa cb-embedding of BMOQ into BMOS under this assumption. According to thecharacterization (1.2), it holds for Markov metrics on commutative spaces (Ω, µ).

1.4. The Euclidean metric. Before using Markov metrics in our approach toCalderon-Zygmund theory, it is illustrative to recover the Euclidean metric from asuitable Markov semigroup. Let S = (Ht)t≥0 denote the classical heat semigroupon Rn, with kernels

ht(x, y) =1

(4πt)n2exp

(−|x− y|2

4t

).

Take Q =(Rj,t, σj,t, γj,t) : (j, t) ∈ Z+ × R+

determined by

• σ2j,t ≡ 2e√

πj

n2 e−j and γ2j,t ≡ j

n2 ≥ 1,

• Rj,tf(x) =1

|B√4jt(x)|

∫B√

4jt(x)

f(y) dy.

Note that σj,t and γj,t are allowed to be essentially bounded functions in Rn, butin this case it suffices to take constant functions. In the definition of Rj,t, we writeBr(x) to denote the Euclidean ball in Rn centered at x with radius r. It is clear thatRj,t defines a cpu map on L∞(Rn). To show that Q defines a Markov metric, weneed to check that it provides a Hilbert module majorization of the heat semigroupand the metric integrability condition holds. The latter is straightforward, whilethe Hilbert module majorization reduces to check that

ht(x, y) ≤ 2e√π

∑j≥1

jn2 e−j

|B√4jt(x)|

χB√4jt(x)

(y).


This can be justified by determining the unique corona centered at x with radii√4(j − 1)t and

√4jt where y lives, details are left to the reader. Note that we

could have taken γj,t ≡ 1 and still obtain a Markov metric. Our choice will bejustified below and also in the next section, where we shall need γj,t ≡ j

n2 to

compare BMOQ with other BMO spaces which interpolate. Before that, our onlyevidences that this is the right Markov metric in the Euclidean case are the factthat the Rj,t’s are averages over Euclidean balls and the isomorphism

BMOQ = BMORn ,

where the latter space is the usual BMO space in Rn

∥f∥BMORn = supB⊂Rn

( 1

|B|

∫B

∣∣f(x)− fB∣∣2 dx) 1

2

.

Here, the supremum is taken over all Euclidean balls B in Rn and fB stands for theaverage of f over B. Let us justify this isomorphism. If we pick Mtf(x) = R1,tf(x)it follows from a standard calculation that∣∣Rj,tf(x)−Mtf(x)

∣∣2(1.3)

=∣∣∣ 1

|B√4t(x)|

∫B√

4t(x)

(f(y)− fB√

4jt(x)

)dy∣∣∣2

≤ jn2

|B√4jt(x)|

∫B√

4jt(x)

∣∣f(y)− fB√4jt(x)

∣∣2 dy = jn2

(Rj,t|f |2 − |Rj,tf |2

)(x).

This automatically yields the following inequality

∥f∥2BMOQ≲ sup

j,tess supx∈Rn

1

|B√4jt(x)|

∫B√

4jt(x)

∣∣f(y)− fB√4jt(x)

∣∣2 dy ≤ ∥f∥2BMORn.

The converse is even simpler, since taking j = 1 we obtain

∥f∥2BMORn= sup

t>0ess supx∈Rn

1

|B√4t(x)|

∫B√

4t(x)

∣∣f(y)− fB√4t(x)

∣∣2 dy= sup

t>0

∥∥∥γ−11,t

[R1,t|f |2 − |R1,tf |2

]γ−11,t

∥∥∥L∞(Rn)

≤ ∥f∥2BMOQ.

Remark 1.6. The term |Rj,tf −Mtf | did not play a significant role at this point.More generally, the above argument also works for any doubling metric space Ωequipped with a Borel measure µ: µ(B(x, 2r)) ≤ Cµ(B(x, r)) for every x ∈ Ω andr > 0, with B(x, r) = y ∈ Ω : dist(x, y) ≤ r. As we shall see later, the additionalterm |Rj,tf −Mtf | in the BMOQ-norm appears to include Tolsa’s RBMO spaces[62] in those measure spaces (Ω, µ) for which we can find an appropriate Dirichletform which provides us with a Markov semigroup acting on (Ω, µ).

Remark 1.7. A related semigroup BMO norm is

∥f∥BMOcS= sup

t≥0

∥∥∥(Ht

[|f − Htf |2

]) 12∥∥∥∞.

All the norms consider so far are equivalent for the heat semigroup S = (Ht)t≥0 onRn, generated by the Laplacian ∆ =

∑nj=1 ∂

2xj. In fact, we may also consider by

subordination the Poisson semigroup P = (Pt)t≥0 on Rn generated by the square


root√−∆, or even other subordinations [23] . Then, elementary calculations give

the following norm equivalences up to dimensional constants

∥f∥BMORn ∼ ∥f∥BMOP ∼ ∥f∥BMOP ∼ ∥f∥BMOS ∼ ∥f∥BMOS ∼ ∥f∥BMOQ .

Moreover, let R = L∞(Rn)⊗M denote the von Neumann algebra tensor product ofL∞(Rn) with a noncommutative measure space (M, τ). Define the norm in BMORas ∥f∥BMOR = max∥f∥BMOc

R, ∥f∗∥BMOc

R, where

∥f∥BMOcR

= supB balls

∥∥∥( 1

|B|

∫B

∣∣f(x)− fB∣∣2dx) 1

2∥∥∥M.

Then, the same norm equivalences hold in the semicommutative case

∥f∥BMOR ∼ ∥f∥BMOP⊗∼ ∥f∥BMOP⊗

∼ ∥f∥BMOS⊗∼ ∥f∥BMOS⊗

,

where S⊗,t = St ⊗ idM and P⊗,t = Pt ⊗ idM. Moreover, by Remark 1.5, all thesenorms are in turn equivalent to the norm in BMOQR , with the Markov metricwhich arises tensorizing the canonical one with the identity/unit of M.

2. Algebraic CZ theory

In classical Calderon-Zygmund theory, Lp boundedness of CZOs follows fromL2 boundedness under a smoothness condition on the kernel. Our next goal is toidentify which are the analogues of these conditions for semifinite von Neumannalgebras equipped with a Markov metric, and to show Lp boundedness of CZOsfulfilling them. Our new conditions are certainly surprising. The boundednessfor p = 2 must be replaced by a certain mixed-norm estimate (which reduces inthe classical theory to L2 boundedness), while Hormander kernel smoothness willbe formulated intrinsically without any reference to the kernel. These abstractassumptions will adopt a more familiar form in the specific cases that we shallconsider in the forthcoming sections.

In order to give a Calderon-Zygmund framework for von Neumann algebras westart with some initial considerations, which determine the general form of Markovmetrics that we shall work with. Consider a Markov metric Q associated to aMarkov semigroup S = (St)t≥0 acting on (M, τ). Then, we shall assume that thecpu maps Rj,t from Q are of the following form

(2.1)M ρj−→ Nρ

Eρ−→ ρ1(M) ≃ M,

Rj,tf = Eρ(qj,t)− 1

2Eρ

(qj,tρ2(f)qj,t

)Eρ(qj,t)

− 12 ,

where ρ1, ρ2 : M → Nρ are ∗-homomorphisms into certain von Neumann algebraNρ, the map Eρ : Nρ → ρ1(M) is an operator-valued weight and the qj,t’s areprojections in Nρ. In particular, we shall assume that our formula for Rj,tf makessense so that qj,t and qj,tρ2(f)qj,t belong to the domain of Eρ, see Section 2.1 forfurther details. Our model provides a quite general form of Markov metric whichincludes the Markov metric for the heat semigroup considered before. Indeed, takeNρ = L∞(Rn × Rn) with ρ1f(x, y) = f(x) and ρ2f(x, y) = f(y). Let Eρ be theintegral in Rn with respect to the variable y and set

qj,t(x, y) = χB√4jt(x)

(y) = χB√4jt(y)

(x) = χ|x−y|<√4jt.


Then, it is straightforward to check that we recover from (2.1) the Rj,t’s for theheat semigroup. Note that the qj,t(x, ·)’s reproduce in this case all the Euclideanballs in Rn. Morally, this is why we call Q a Markov metric, since it codifies somesort of underlying metric in (M, τ). According to our definition of BMOQ, we shallalso consider projections qt in Nρ and cpu maps

(2.2) Mtf = Eρ(qt)− 1

2Eρ

(qtρ2(f)qt

)Eρ(qt)

− 12 .

2.1. Operator-valued weights. In this subsection we briefly review the definitionand basic properties of operator-valued weights from [26, 27]. A unital, weaklyclosed ∗-subalgebra is called a von Neumann subalgebra. A conditional expectationEM : N → M onto a von Neumann subalgebra M is a positive unital projectionsatisfying the bimodular property EM(a1f a2) = a1EM(f)a2 for all a1, a2 ∈ M. Itis called normal if supα EM(fα) = EM(supα fα) for bounded increasing nets (fα)in N+. A normal conditional expectation exists if and only if the restriction of τto the von Neumann subalgebra M remains semifinite [61]. Any such conditionalexpectation is trace preserving τ EM = τ .

The extended positive part M+ of the von Neumann algebra M is the set oflower semicontinuous maps m : M∗,+ → [0,∞] which are linear on the positivecone, m(λ1ϕ1 + λ2ϕ2) = λ1m(ϕ1) + λ2m(ϕ2) for λj ≥ 0 and ϕj ∈ M∗,+. Theextended positive part is closed under addition, increasing limits and is fixed bythe map x 7→ a∗xa for any a ∈ M. It is clear that M+ sits in the extended positivepart. When M is abelian, we find M ≃ L∞(Ω, µ) for some measure space (Ω, µ)and the extended positive part corresponds in this case to the set of µ-measurablefunctions on Ω (module sets of zero measure) with values in [0,∞]. A hardercharacterization of the extended positive part for arbitrary von Neumann algebraswas found by Haagerup in [26]. Assume that M acts on H and consider a positiveoperator A affiliated with M with spectral resolution A =

∫R+λdeλ. Then, we may

construct an associated element in M+

mA(ϕ) =

∫R+

λd(ϕ(eλ)).

In general, any m ∈ M+ has a unique spectral resolution

m(ϕ) =

∫R+

λd(ϕ(eλ)) +∞ϕ(p)

where the eλ’s form an increasing family of projections in M and p is the projection1M− limλ eλ. Moreover, the map λ 7→ eλ is strongly continuous from the right andwe find that e0 = 0 iff m does not vanish on M+

∗ \ 0, while p = 0 iff the familyof ϕ ∈ M+

∗ with m(ϕ) <∞ is dense in M+∗ .

Operator-valued weights appear as “unbounded conditional expectations” andthe simplest nontrivial model is perhaps a partial trace EM = trA ⊗ idM withN = A⊗M and A a semifinite non-finite von Neumann algebra. In general, anoperator-valued weight from N to M is just a linear map

EM : N+ → M+ satisfying EM(a∗fa) = a∗EM(f)a

for all a ∈ M. As usual, EM is called normal when supα EM(fα) = EM(supα fα) for

bounded increasing nets (fα) in N+. Since a∗fb = 1

4

∑3k=0 i

−k(a+ ikb)∗f(a+ ikb)


by polarization, we see that bimodularity of conditional expectations is equivalentto EM(a∗fa) = a∗EM(f)a for a ∈ M. In particular, the fundamental propertieswhich operator-valued weights loose with respect to conditional expectations areunitality and the fact that unboundedness is allowed for the image. Additionally,when M = C the map EM becomes an ordinary weight on N . In analogy withordinary weights, we take

Lc∞(N ;EM) =

f ∈ N :

∥∥EM(f∗f)∥∥M <∞

.

Note that when EM = trA ⊗ idM with N = A⊗M, Lc∞(N ;EM) are the Hilbert

space valued noncommutative L∞ spaces defined in [30], which we denote byLc2(A)⊗M. Let NEM be the linear span of f∗1 f2 with f1, f2 ∈ Lc

∞(N ;EM). Thenwe find

i) NEM = spanf ∈ N+ : ∥EMf∥ <∞,ii) Lc

∞(N ;EM) and NEM are two-sided modules over M,

iii) EM has a unique linear extension EM : NEM → M satisfying

EM(a1fa2) = a1EM(f)a2 with f ∈ NEM and a1, a2 ∈ M.

In particular, if EM(1) = 1 we recover a conditional expectation onto M. Anoperator-valued weight EM is called faithful if EM(f∗f) = 0 implies f = 0 andsemifinite when Lc

∞(N ;EM) is σ-weakly dense in N . It is of interest to determinefor which pairs (N ,M) we may construct n.s.f. operator-valued weights. Amongother results, Haagerup proved in [27] that this is the case when both von Neumannalgebras are semifinite and there exists a unique trace preserving one. Note thatconditional expectations do not always exist in this case. He also proved that givenEMj

n.s.f. operator-valued weights in (Nj ,Mj) for j = 1, 2, there exists a uniquen.s.f. operator-valued weight EM1⊗M2

associated to (N1⊗N2,M1⊗M2) such that(ϕ1 ⊗ ϕ2) EM1⊗M2

= (ϕ1 EM1) ⊗ (ϕ2 EM2

) for any pair (ϕ1, ϕ2) of normalsemifinite faithful weights on (M1,M2).

2.2. Algebraic/analytic conditions. The identity

Tf(x) =

∫Ω

k(x, y)f(y) dµ(y)

is just a vague expression to consider classical Calderon-Zygmund operators. It iswell-known that particular realizations as above are only meaningful outside thesupport of f and understanding k as a distribution which coincides with a locallyintegrable function on Rn×Rn \∆. Instead of that, we shall not specify any kernelrepresentation of our CZOs since our conditions below will be formulated in a moreintrinsic way. These kernel representations will appear later on in this paper withthe concrete examples that we shall consider.

Let T be a densely defined operator on M, which means that Tf ∈ M for allf in a weak-∗ dense subalgebra AM of M. Our assumption does not necessarilyhold for classical Calderon-Zygmund operators defined in abelian von Neumannalgebras (M, τ) = L∞(Ω, µ), but it is true for the truncated singular integraloperators satisfying the standard size condition for the kernel, take for instanceAM = M∩L1(M). In particular, this is not a crucial restriction since we shall beable to take Lp-limits as far as our estimates below are independent of T . Our aim is


to settle conditions on T of CZ type assuring that T : L∞(M) → BMOcQ, provided

(M, τ) comes equipped with a Markov metric Q. In this paragraph, we establishsome preliminary algebraic and analytic conditions on the Markov metric and theCZO. Consider ∗-homomorphisms π1, π2 : M → Nπ and an operator-valued weightEπ : Nπ → π1(M) which may or may not coincide with ρ1, ρ2 and Eρ from (2.1).Assume there exists a (densely defined) map

(2.3)T : ANπ ⊂ Nπ → Nρ

satisfying T π2 = ρ2 T on AM.

Algebraic conditions:

i) Q-monotonicity of Eρ

Eρ(qj,t|ξ|2qj,t) ≤ Eρ(|ξ|2)for all ξ ∈ Nρ and every projection qj,t determined by Q via the identity in(2.1). Similarly, we assume the same inequality holds when we replace theqj,t’s by the qt’s appearing in (2.2).

ii) Right B-modularity of T

T(η π1ρ

−11 (b)

)= T (η)b

for all η ∈ ANπand all b lying in some von Neumann subalgebra B of

ρ1(M) which includes Eρ(qt), Eρ(qj,t) and ρ1(γj,t) for every projection qtand qj,t determined by Q via the identities in (2.1) and (2.2).

As we shall see both conditions trivially hold in the classical theory, where thefirst condition essentially says that integrating a positive function over a “Markovmetric ball” is always smaller than integrating it over the whole space, while thesecond condition allows to take out x-dependent functions from the y-dependentintegral defining T . Our conditions remain true in many other situations, which willbe explored below in this paper. Nevertheless, condition i) suggests that certainamount of commutativity might be necessary to work with Markov metrics.

To state our analytic conditions we introduce an additional von Neumann algebraNσ containing Nρ as a von Neumann subalgebra. Then, we consider derivationsδ : Nρ → Nσ given by the difference δ = σ1 − σ2 of two ∗-homomorphisms, sothat δ(ab) = σ1(a)σ1(b)− σ2(a)σ2(b) = δ(a)σ1(b) + σ2(a)δ(b) as expected. We alsoconsider the natural amplification maps

Rj,t : Nρ ∋ ξ 7→ Eρ(qj,t)− 1

2Eρ(qj,tξqj,t)Eρ(qj,t)− 1

2 ∈ ρ1(M),

Mt : Nρ ∋ ξ 7→ Eρ(qt)− 1

2Eρ(qtξqt)Eρ(qt)− 1

2 ∈ ρ1(M).

Analytic conditions:

i) Mean differences conditions

• Rj,t(ξ∗ξ)− Rj,t(ξ)

∗Rj,t(ξ) ≤ Φj,t

(δ(ξ)∗δ(ξ)

),

•[Rj,t(ξ)− Mt(ξ)

]∗[Rj,t(ξ)− Mt(ξ)

]≤ Ψj,t

(δ(ξ)∗δ(ξ)

),

for some derivation δ : Nρ → Nσ and cpu maps Φj,t,Ψj,t : Nσ → ρ1(M).

ii) Metric/measure growth conditions


• 1 ≤ π1ρ−11 Eρ(qt)

− 12Eπ(a

∗tat)π1ρ

−11 Eρ(qt)

− 12 ≲ π1ρ

−11 (γ2j,t),

• 1 ≤ π1ρ−11 Eρ(qj,t)

− 12Eπ(a

∗j,taj,t)π1ρ

−11 Eρ(qj,t)

− 12 ≲ π1ρ

−11 (γ2j,t),

for some family of operators at, aj,t ∈ Nπ to be determined later on.

A complete determination of the operators at and aj,t is only possible after imposingadditional size and smoothness conditions in our definition of Calderon-Zygmundoperator below. Nevertheless, we shall see that these operators will play the roleof “dilated Markov balls” as it is the case in classical CZ theory. In fact, in theclassical case our last condition trivially holds for doubling measures, and also formeasures of polynomial or even exponential growth provided we find a Markovmetric with large enough γj,t’s. Our assertions will be illustrated below. The firstcondition takes the form in the classical case of a couple of easy consequences ofJensen’s inequality, namely

(2.4)

−∫B1

|f |2dµ−∣∣∣−∫

B1

fdµ∣∣∣2 ≤ −

∫B1×B1

∣∣f(y)− f(z)∣∣2dµ(y)dµ(z),∣∣∣ −

∫B1

fdµ − −∫B2

fdµ∣∣∣2 ≤ −

∫B1×B2

∣∣f(y)− f(z)∣∣2dµ(y)dµ(z).

2.3. CZ extrapolation. Now we introduce CZOs in this context. As we alreadymentioned, we consider a priori densely defined (unbounded) maps T : AM → Mwhose amplified maps are right B-modules according to our algebraic assumptionsabove. In addition, we impose three conditions generalizing L2 boundedness, thesize and the smoothness conditions for the kernel.

Calderon-Zygmund type conditions:

i) Boundedness condition

T : Lc∞(Nπ;Eπ) → Lc

∞(Nρ;Eρ).

ii) Size “kernel” condition

• Mt

(∣∣T (π2(f)(Aj,t − at))∣∣2) ≲ γ2j,t∥f∥2∞,

• Rj,t

(∣∣T (π2(f)(Aj,t − aj,t))∣∣2) ≲ γ2j,t∥f∥2∞,

for a family of operators Aj,t ∈ Nπ with Aj,t ≥ aj,t, at to be determined.

iii) Smoothness “kernel” condition

• Φj,t

(∣∣δ(T (π2(f)(1 − aj,t)))∣∣2) ≲ γ2j,t∥f∥2∞,

• Ψj,t

(∣∣δ(T (π2(f)(1−Aj,t)))∣∣2) ≲ γ2j,t∥f∥2∞.

Let T : AM → M be a densely defined map which admits an amplification Tsatisfying (2.3). Any such T will be called an algebraic column CZO whenever theamplification map is right B-modular and satisfies the CZ conditions we have givenabove. At first sight, our boundedness assumption might appear to be unrelatedto the classical condition. The reader could have expected the L2 boundednessof T , but our assumption is formally equivalent to it in the classical case andgives the right condition for more general algebras. On the other hand, our size


and smoothness conditions are intrinsic in the sense that the kernel is not specifiedunder this degree of generality. We shall recover classical kernel type estimates fromour conditions in our examples below. As explained above, the operators at, aj,tand Aj,t play the role of dilated Markov balls and our conditions were somehowmodeled by Tolsa’s arguments in [62]. Perhaps a significant difference —in contrastto Tolsa’s approach— is that our smoothness condition is analog to a Hormandertype condition, more than the (stronger) Lipschitz regularity assumption.

Theorem 2.1. Let (M, τ) be a noncommutative measure space equipped with aMarkov semigroup S = (St)t≥0 with associated Markov metric Q which fulfills ouralgebraic and analytic assumptions. Then, any algebraic column CZO T defines abounded operator

T : AM → BMOcQ.

Proof. The first goal is to estimate the norm of

A = γ−1j,t

(Rj,t|Tf |2 − |Rj,tTf |2

)γ−1j,t .

The map Πj,t : a ⊗ b ∈ M⊗Rj,tM 7→ 1 ⊗ Rj,t(a)b ∈ 1 ⊗ M extends to a right(1 ⊗ M)-module projection, which is well-defined in the sense that ⟨ξ, ξ⟩Rj,t = 0implies Πj,t(ξ) = 0. Now, since

A = γ−1j,t

⟨Tf ⊗ 1− 1⊗Rj,tTf , Tf ⊗ 1− 1⊗Rj,tTf

⟩Rj,t

γ−1j,t ,

we may use Πj,t to deduce the following identity

A =⟨(id−Πj,t)(Tf ⊗ γ−1

j,t ) , (id−Πj,t)(Tf ⊗ γ−1j,t )⟩Rj,t

.

Consider the amplification maps Rj,t and Πj,t determined by

Rj,t = Rj,t ρ2 and Πj,t = Πj,t (ρ2 ⊗ id).

By (2.3), it turns out that A = ⟨a,a⟩Rj,twhere

a = (id− Πj,t)(ρ2Tf ⊗ γ−1j,t )

= (id− Πj,t)(T π2f ⊗ γ−1j,t )

= (id− Πj,t)(T (π2(f)aj,t)⊗ γ−1

j,t

)+ (id− Πj,t)(T

(π2(f)(1− aj,t))⊗ γ−1

j,t

)= a1 + a2

According to Lemma 1.1 i), we may estimate A as follows

A ≲⟨a1,a1

⟩Rj,t

+⟨a2,a2

⟩Rj,t

= A1 +A2.

Since Πj,t(n⊗ b) = 1⊗ Rj,t(n)b, the Kadison-Schwarz inequality yields⟨Πj,t(n⊗ b), Πj,t(n⊗ b)

⟩Rj,t

≲⟨n⊗ b, n⊗ b

⟩Rj,t

.

In conjunction with Lemma 1.1 i) again, we deduce the following estimate for A1

A1≲⟨T (π2(f)aj,t)⊗ γ−1

j,t , T (π2(f)aj,t)⊗ γ−1j,t

⟩Rj,t

= γ−1j,t Rj,t

(∣∣T (π2(f)aj,t)∣∣2)γ−1j,t .

In order to bound the term in the right hand side, we apply (2.1) and the propertiesof the operator-valued weight Eρ together with our algebraic conditions. Indeed, wefirst use theQ-monotonicity of Eρ; then the fact that it commutes with the left/right


multiplication by elements affiliated to M (like γ−1j,t or Eρ(qj,t)

−1/2); finally we usethe right B-modularity of the amplification of T :

γ−1j,t Rj,t

(∣∣T (π2(f)aj,t)∣∣2)γ−1j,t

≤ γ−1j,t Eρ(qj,t)

− 12Eρ

(∣∣T (π2(f)aj,t)∣∣2)Eρ(qj,t)− 1

2 γ−1j,t

= Eρ

(γ−1j,t Eρ(qj,t)

− 12

∣∣T (π2(f)aj,t)∣∣2Eρ(qj,t)− 1

2 γ−1j,t

)= Eρ

∣∣∣T(π2(f)aj,t π1ρ−11

(Eρ(qj,t)

− 12 γ−1

j,t

)︸︷︷︸ξ1

)∣∣∣2 = Eρ|T (ξ1)|2.

Now, our first CZ condition i) gives the boundedness we need since

∥A1∥M ≤∥∥T (ξ1)∥∥2Lc

∞(Nρ;Eρ)≲∥∥ξ1∥∥2Lc

∞(Nπ ;Eπ)

=∥∥∥Eπ

(∣∣π2(f)aj,t π1ρ−11

(Eρ(qj,t)

− 12 γ−1

j,t

)∣∣2)∥∥∥M

≤∥∥∥π1ρ−1

1

(γ−1j,t Eρ(qj,t)

− 12

)∗Eπ(a

∗j,taj,t)π1ρ

−11

(Eρ(qj,t)

− 12 γ−1

j,t

)∥∥∥M∥f∥2∞.

The last term on the right is dominated by ∥f∥2∞ according to our second analyticcondition on metric/measure growth. The estimate for A2 is simpler. Indeed, if we

set ξ2 = T (π2(f)(1− aj,t)) then

A2 =⟨(id− Πj,t)(ξ2 ⊗ γ−1

j,t ) , (id− Πj,t)(ξ2 ⊗ γ−1j,t )⟩Rj,t

= γ−1j,t

(Rj,t|ξ2|2 −

∣∣Rj,t(ξ2)∣∣2)γ−1

j,t ≤ γ−1j,t Φj,t

(|δξ2|2

)γ−1j,t ≲ ∥f∥2∞1,

where the first inequality holds for some derivation δ : Nρ → Nσ and some cpu mapΦj,t : Nσ → ρ1(M) by our first analytic condition on mean differences. Then ourCZ condition iii) on kernel smoothness justifies our last estimate. Our estimates sofar prove the desired estimate

supt>0

supj≥1

∥∥∥(γ−1j,t

[Rj,t|Tf |2 − |Rj,tTf |2

]γ−1j,t

) 12∥∥∥M

≲ ∥f∥∞.

Therefore, it remains to estimate the norm of

B = γ−1j,t

(∣∣Rj,tTf −MtTf∣∣2)γ−1

j,t .

To do so, we decompose the middle term using (2.3) as follows

Rj,tTf −MtTf

= Rj,t(ρ2Tf)− Mt(ρ2Tf)

= Rj,t

(T(π2(f)aj,t

))− Mt

(T(π2(f)at

))+

[Rj,t

(T(π2(f)(1− aj,t)

))− Mt

(T(π2(f)(1− at)

))]= b1 − b2 + b3.

Letting Bj = γ−1j,t |bj |2γ−1

j,t we get B ≲ B1 +B2 +B3. By Kadison-Schwarz we get

B1 ≤ γ−1j,t Rj,t

(∣∣T (π2(f)aj,t)∣∣2)γ−1j,t ≲ ∥f∥2∞,


where the last inequality was justified in our estimate of A1 above. Replacing qj,t byqt, the same argument serves to control the term B2. To estimate B3 we decomposeb3 as follows

b3 =[Rj,t

(T(π2(f)(1−Aj,t)

))− Mt

(T(π2(f)(1−Aj,t)

))]+ Rj,t

(T(π2(f)(Aj,t − aj,t)

))− Mt

(T(π2(f)(Aj,t − at)

))= b31 + b32 − b33.

Taking ξ3 = T(π2(f)(1 − Aj,t)

)and applying our analytic condition i) on mean

differences together with our CZ condition iii) on kernel smoothness, we obtain that

γ−1j,t |b31|2γ−1

j,t ≲ γ−1j,t Ψj,t

(|δξ3|2

)γ−1j,t ≲ ∥f∥2∞.

It remains to estimate the terms B32 and B33. Applying the Kadison-Schwarzinequality, it is easily checked that these terms are also dominated by ∥f∥2∞ bymeans of our CZ size kernel condition ii). Altogether, we have justified that

supt>0

infMtcpu

supj≥1

∥∥∥(γ−1j,t

[|Rj,tf −Mtf |2

]γ−1j,t

) 12∥∥∥M

≲ ∥f∥∞.

Combining our estimates for A and B, we deduce that T : AM → BMOcQ.

TheAM → BMOrQ boundedness of the map T is equivalent to theAM → BMOc

Qboundedness of the map T †(f) = T (f∗)∗. According to this, an algebraic CZO isany column CZO T for which T † remains a column CZO. By Theorem 2.1, weknow that any algebraic CZO T associated to (M, τ,Q) as above is automaticallyAM → BMOQ bounded. Assuming L2 boundedness and regularity of the Markovsemigroup, we may interpolate via Theorem 1.4. Under the same assumptions forT ∗, we may also dualize and obtain the following extrapolation result.

Corollary 2.2. Let (M, τ) be a noncommutative measure space equipped with aMarkov regular semigroup S = (St)t≥0 and a Markov metric Q = (Rj,t, σj,t, γj,t)fulfilling our algebraic and analytic assumptions. Then, every L2-bounded algebraicCZO T satisfies that JpT : Lp(M) → L

p(M) for p > 2. Applying duality, similarconditions for T ∗ yield Lp-boundedness of TJp for every 1 < p < 2.

Remark 2.3. Theorem 2.1 admits a completely bounded version in the category ofoperator spaces. Since the operator space structure [22, 52] of BMO is determinedby

Mm(BMOS(M)) = BMOS(Mm(M))

for S = (idMm ⊗St)t≥0, we just need to replace M byMm(M) everywhere, amplifyall the involved maps by tensorizing with idMm

and require that the hypotheseshold with constants independent of m. Then, we obtain the cb-boundedness of T .

Remark 2.4. As noticed in the Introduction, a common scenario is given by thechoice Nρ = M⊗M with ρ1(f) = f⊗1 and ρ2(f) = 1⊗f , together with Eρ = id⊗τand πj = ρj for j = 1, 2. In this case, it is clear that the amplification map is givenby

T = idM ⊗ T so that T π2 = ρ2T.

In particular, it turns out that the L2 boundedness of T in Corollary 2.2 followsautomatically from our CZ boundedness condition i). This is the case in classicalCalderon-Zygmund theory. It is also true when Nρ = M⊗A for an auxiliaryalgebra A and ρ2 = flipσ, where σ : M → A⊗M is a ∗-homomorphism satisfying


Eρ ρ2(f) = τ(f)1M. This leads to another significant family of examples. It ishowever surprising that in general, the L2 boundedness and the CZ boundednessassumptions are a priori unrelated. Thus, CZ extrapolation requires in this contextto verify two boundedness conditions. It would be quite interesting to explore thecorresponding “T (1) problems” that arise naturally.

2.4. The classical theory revisited. We now illustrate our algebraic approachin the classical context of Euclidean spaces with the Lebesgue measure. This willhelp us to understand some of our conditions and will show how some others areautomatic in a commutative framework. Take M = L∞(Rn) with the Lebesguemeasure and S = (Ht)t≥0 the heat semigroup Ht = exp(t∆). In Paragraph 1.4 weintroduced the Markov metric Q given by

• σ2j,t ≡ 2e√

πj

n2 e−k and γ2j,t ≡ j

n2 ≥ 1,

• Rj,tf(x) =1

|B√4jt(x)|

∫B√

4jt(x)

f(y) dy.

Moreover, as we explained at the beginning of this section

Rj,tf = Eρ(qj,t)− 1

2Eρ

(qj,tρ2(f)qj,t

)Eρ(qj,t)

− 12

satisfies our basic assumption (2.1). Here the amplification von Neumann algebrais Nρ = L∞(Rn × Rn), the ∗-homomorphisms ρjf(x1, x2) = f(xj), the projectionsqj,t(x, y) = χB√

4jt(x)(y), and the operator-valued weight Eρ is the integration map

with respect to the second variable. The cpu mapMt which appears in the definitionof BMOQ is still taken by Mt = R1,t as in Paragraph 1.4.

Taking Nπ = Nρ and T a standard CZO in Rn, the algebraic conditions triviallyhold in this case. Let Ex

j,k,t = B√4jt(x)×B√

4kt(x). Taking Φj,t and Ψj,t to be theaveraging maps over Ex

j,j,t and Exj,1,t respectively and the family of dilated balls

(Aj,t(x, y), aj,t(x, y)) = (χαB√4jt(x)

(y), χ5B√4jt(x)

(y)) with a ≥ 5, we may recover

the conditions as we explained right after stating them. Let us now show how ouralgebraic CZ conditions hold from the classical ones. The boundedness conditionreduces to the classical one, see Remark 4.2 A). Our size conditions can be rewrittenas follows:

• ess supx∈Rn

−∫B√

4t(x)

∣∣∣ ∫aB√

4jt(x)\5B√4t(x)

k(y, z)f(z)dz∣∣∣2dy ≲ j

n2 ∥f∥2∞,

• ess supx∈Rn

−∫B√

4jt(x)

∣∣∣ ∫aB√

4jt(x)\5B√4jt(x)

k(y, z)f(z)dz∣∣∣2dy ≲ j

n2 ∥f∥2∞.

The above conditions follow from the usual size condition

|k(y, z)| ≲ 1

|z − y|n.

Next, taking Exj,k,t as above, our smoothness conditions are:

• ess supx∈Rn

−∫Ex

j,1,t

(∫(5B√

4jt(x))c

(k(y1, z)− k(y2, z)

)f(z)dz

)2dy1dy2 ≲ j

n2 ∥f∥2∞,

• ess supx∈Rn

−∫Ex

j,j,t

(∫(aB√

4jt(x))c

(k(y1, z)− k(y2, z)

)f(z)dz

)2dy1dy2 ≲ j

n2 ∥f∥2∞.


The above conditions easily follow from the usual Hormander condition

ess supy1,y2∈Rn

∫|y1−z|≥2|y1−y2|

|k(y1, z)− k(y2, z)| dz <∞.

Note that our algebraic CZ conditions are slightly weaker than the classical CZconditions, but still sufficient to provide L∞ → BMO boundedness of the CZ map.

Remark 2.5. Our size condition is only used to estimate B in the proof of Theorem2.1. We saw in Paragraph 1.4 that B ≲ A for the Euclidean metric. Thus, our sizecondition is not necessary here, as it also happens in the classical formulation.

3. Applications I — Commutative spaces

In this section we give specific constructions of Markov metrics on two basiccommutative spaces: Riemannian manifolds with nonnegative Ricci curvature andGaussian measure spaces. Beyond the Euclidean-Lebesguean setting consideredabove, these are the most relevant settings over which Calderon-Zygmund theoryhas been studied. As a good illustration of our algebraic method, we shall recoverthe extrapolation results. Noncommutative spaces will be explored later on.

3.1. Riemannian manifolds. Let (Ω, µ) be a measure space equipped with aMarkov semigroup, so that we may construct the corresponding semigroup typeBMO space. In order to study the L∞ → BMO boundedness of CZOs in (Ω, µ) itis essential to identify a Markov metric to work with. Now we provide sufficientconditions for a semigroup on a Riemannian manifold to yield a Markov metricsatisfying our algebraic/analytic conditions, so that Theorem 2.1 is applicable. Letus consider an n-dimensional complete Riemannian manifold (M, g) equipped withthe geodesic distance d determined by the Riemannian metric g. Denote the volumeof a geodesic ball centered at x with radius r by volg(Br(x)). Let SM be a Markovsemigroup on M given by

SM,tf(x) =

∫M

st(x, y)f(y) dy.

Proposition 3.1. Assume that

i) M has Ricci curvature ≥ 0.

ii) The kernel admits an upper bound

st(x, y) ≲ϕ(t)n+ε

volg(Bϕ(t)(x))(d(x, y) + ϕ(t))n+ε,

for some strictly positive function ϕ and some parameter ε > 0.

Then SM admits a Markov metric satisfying the algebraic/analytic conditions.

Proof. If Σj,t(x) = B2jϕ(t)(x), our assumption. gives

(3.1) st(x, y) ≲∞∑j=1

2−j(n+ε)

volg(Σ0,t(x))χΣj,t(x)(y).

According to Davies [17, Theorem 5.5.1], non-negative Ricci curvature implies

volg(Br(x)) ≤ cnrn,


volg(Bγr(x)) ≤ γnvolg(Br(x))

for all x ∈M , r > 0 and γ > 1. In particular, volg(Σj,t(x)) ≤ 2jnvolg(Σ0,t(x)). By

(3.1), this implies that (σj,t, γj,t) = (2−jε/2, 1) forms a Markov metric for SM inconjunction with the averaging maps

Rj,tf(x) = −∫Σj,t(x)

f(y)dy for (j, t) ∈ Z+ × R+.

Our construction for M = L∞(M) follows the basic model in the Introductionand the one used above in the Euclidean setting: Nρ = Nπ = M⊗M with ρjthe canonical inclusion maps and qj,t(x, y) = χΣj,t(x)(y) = χΣj,t(y)(x). Then, thealgebraic conditions for the Markov metric are obviously satisfied. Let us now checkthe analytic conditions. Taking Nσ = M⊗M⊗M, the derivation δ : Nρ → Nσ

given by δ(a ⊗ b) = a ⊗ (1 ⊗ b − b ⊗ 1) and the maps Mt = R1,t, it turns outthat the mean difference conditions follow from Jensen’s inequality on normalizedballs of (M, g) as it follows from our comments after the definition of the analyticconditions. It remains to consider the metric/measure growth conditions. By takingaj,t(x, y) = χΣj+1,t,(x)(y) and (qt, at) = (q1,t, a1,t), these conditions reduce to showthat

volg(B2j+1ϕ(t)(x)) ≈ volg(B2jϕ(t)(x)).

This follows in turn from the fact that M has a non-negative Ricci curvature.

Let (M, g) be a complete Riemannian manifold with non-negative Ricci curvatureand let ∆ be the Laplace-Beltrami operator. The heat semigroup S∆ generated by∆ admits a kernel on (M, g) satisfying the upper bound estimate mentioned in theabove proposition. We know from Davies [17, Theorem 5.5.11] that the heat kernelsatisfies

(3.2) ht(x, y) ≤aδ

volg(B√t(x))

exp(− d(x, y)2

4(1 + δ)t

)for any δ > 0 and certain constant aδ. This implies that

ht(x, y) ≲ aδvolg(B√

t(x))

(4(1 + δ)t)n+ε2

(d(x, y)2 + 4(1 + δ)t)n+ε2

≲ (√

4(1 + δ)t)n+ε

volg(B√4(1+δ)t(x))(d(x, y) +

√4(1 + δ)t)n+ε

,

which gives the expected upper bound with ϕ(t) =√

4(1 + δ)t.

Remark 3.2. Once we have confirmed that algebraic and analytic conditions holdfor the Markov process generated by the Laplace-Beltrami operator ∆, it should benoticed that our CZ conditions are again implied by the classical ones. Arguing as inRemarks 1.6 and 2.5, we see that the Ricci curvature assumption allows us to ignoreour size kernel conditions. Next, it is straightforward to check that the boundednesscondition reduces in this case to standard L2-boundedness. Finally, our discussionin section 2.4 shows that our smoothness kernel condition is guaranteed under theclassical Hormander kernel condition. Note in addition that our conditions alsohold in the row case. In particular, classical CZOs in (M, g) become algebraicCZOs. Moreover, the gaussian upper estimate (3.2) indicates that in (M, g) withthe heat semigroup S∆ we have L

p(M) = Lp(M, g) for 1 < p <∞.


By the discussion above, we have all the ingredients to apply Theorem 2.1 andCorollary 2.2. Let us illustrate it for the Riesz transforms on (M, g). Considerthe Riemannian gradient ∇ = (∂1, ∂2, . . . , ∂n) on (M, g). The Riesz transform on(M, g) is formally defined by

R = (Rj) = ∇(−∆)−12 with Rj = ∂j(−∆)−

12 .

Then we may recover Bakry’s theorem [1] using our algebraic approach. Indeed

integration by parts gives ∥|∇f |∥2 = ∥∆ 12 f∥2 which implies L2-boundedness of

Riesz transforms. Moreover, the Hormander condition follows from [10, 41].

Corollary 3.3. Let (M, g) be a complete n-dimensional Riemannian manifold withnon-negative Ricci curvature. Then for all 1 < p < ∞, there exists a constantCp > 0 such that

∥Rjf∥Lp(M,g) ≤ Cp∥f∥Lp(M,g) for all 1 ≤ j ≤ n.

3.2. The Gaussian measure. Now we study the Ornstein-Uhlenbeck semigroupon the Euclidean space equipped with its Gaussian measure. We shall first constructa Markov metric for it. Then, we shall prove that our algebraic/analytic andCalderon-Zygmund conditions hold for the standard CZOs in this setting. Theinfinitesimal generator of the Ornstein-Uhlenbeck semigroup O = (Ot)t≥0 is theoperator

L =∆

2− x · ∇

on (Rn, µ) with dµ(y) = exp(−|y|2)dy. We have

Otf(x) =1

(π − πe−2t)n2

∫Rn

exp(− |e−tx− y|2

1− e−2t

)f(y) dy

=1

(π − πe−2t)n2

∫Rn

exp(|x|2 − |etx− y|2

e2t − 1

)f(y) dµ(y).

First, we establish a useful lemma showing that the local behavior —i.e. forsmall values of t— of the semigroup type BMO norm for the Ornstein-Uhlenbecksemigroup determines it completely.

Lemma 3.4. Given δ > 0, there exists Cδ > 0 such that

supt≥0

∥∥Ot|f |2 − |Otf |2∥∥∞ ≤ Cδ sup

t<δ

∥∥Ot|f |2 − |Otf |2∥∥∞.

Proof. It is easy to check that

(3.3) Otf(x) = Hv(t)f(e−tx),

for v(t) = 14 (1 − e−2t) and the heat semigroup Ht = exp(t∆). Given t > 0 and

f ∈ L∞(Rn), let F (s) = Hs|Ht−sf |2 for 0 ≤ s ≤ t. According to the definition ofHt, we obtain the following identity

∂sF = (∂sHs)|Ht−sf |2 +Hs[(∂sHt−sf)∗(Ht−sf)] +Hs[(Ht−sf)

∗(∂sHt−sf)]

= ∆Hs|Ht−sf |2 −Hs[(∆Ht−sf)∗(Ht−sf)]−Hs[(Ht−sf)

∗(∆Ht−sf)]

= Hs[∆|Ht−sf |2 − (∆Ht−sf)∗(Ht−sf)− (Ht−sf)

∗(∆Ht−sf)]

= 2Hs|∇Ht−sf |2.


Kadison-Schwarz inequality gives for 0 < u < s

Hu|∇Ht−uf |2 = Hu|Hs−u∇xHt−sf |2 ≤ Hs|∇xHt−sf |2

which implies that ∂sF is increasing and F is convex. Rearranging the inequalityF (s) ≤ 1

2 (F (0) + F (2s)), we get H2t|f |2 − |H2tf |2 ≤ 2Ht(Ht|f |2 − |Htf |2) for anyt ≥ 0. Then, the L∞ contractivity of Ht gives

(3.4)∥∥H2kt|f |2 − |H2ktf |2

∥∥∞ ≤ 2k

∥∥Ht|f |2 − |Htf |2∥∥∞.

On the other hand, choosing kδ such that 2kδv(δ) ≥ 14 and applying (3.3) and (3.4)

supt≥0

∥∥Ot|f |2 − |Otf |2∥∥∞ = sup

t< 14

∥∥Ht|f |2 − |Htf |2∥∥∞

≤ supt<2kδv(δ)

∥∥Ht|f |2 − |Htf |2∥∥∞

≤ 2kδ supt<δ

∥∥Ot|f |2 − |Otf |2∥∥∞.

By the lemma above, it suffices to construct a Markov metric for (Ot)t≥0 with

0 < 2t < 118 . Let v =

√e2t − 1 and consider the following family of balls and

coronas in the gaussian space for (j, t) ∈ Z+ × R+

Σj,t(x) = B(etx,√jv) and Ωj,t(x) = Σj,t(x) \ Σj−1,t(x).

Let j0 = j0(x, t) be the smallest possible integer j satisfying that 0 ∈ Σj,t(x).

The case n = 1. If 1 ≤ j < j0, let

D−j,t(x) =

y ∈ Ωj,t(x) : e

t|x| −√jv ≤ |y| ≤ et|x| −

√j − 1v

,

D+j,t(x) =

y ∈ Ωj,t(x) : e

t|x|+√j − 1v ≤ |y| ≤ et|x|+

√jv.

Then, D−j,t(x) ∪D

+j,t(x) = Ωj,t(x) and we get

(3.5) Otf(x) ≲1

v

( ∑ε=±

1≤j<j0

exp(|x|2−j)∫Dε

j,t(x)

fdµ+∑j≥j0

exp(|x|2−j)∫Σj,t(x)

fdµ)

for any positive f ∈ L∞(R, µ). The above estimate indicates the natural candidatesfor the cpu maps Rj,t and σj,t ∈ L∞(R, µ). When 1 ≤ j < j0 and ε = ±, we define

Rj,t,εf(x) =1

µ(Dεj,t(x))

∫Dε

j,t(x)

fdµ and σ2j,t,ε(x) =

1

vexp(|x|2 − j)µ(Dε

j,t(x)).

Note here we need an extra index for Rj,t’s when j < j0. This is consistent withthe assumptions (i) and (ii) in our definition of Markov metric, since we only needthe index-set of Rj,t’s to be countable. On the other hand, if j ≥ j0, we set

Rj,tf(x) =1

µ(Σj,t(x))

∫Σj,t(x)

fdµ and σ2j,t(x) =

1

vexp(|x|2 − j)µ(Σj,t(x)).

In order to find γj,t,ε and γj,t satisfying the metric integrability condition, we needto estimate µ(Dε

j,t(x)) and µ(Σj,t(x)) respectively. Since the density function µ ismonotone on Dε

j,t(x) we get

µ(D−j,t(x)) =

∫D−

j,t(x)

e−y2

dy ≤ exp(−∣∣et|x| −√jv∣∣2) v√

j,(3.6)


µ(D+j,t(x)) =

∫D+

j,t(x)

e−y2

dy ≤ exp(−∣∣et|x|+√j − 1v

∣∣2) v√j.(3.7)

When j ≥ j0 we use the trivial estimate

µ(Σj,t(x)) =

∫Σj,t(x)

e−y2

dy ≤ 2√jv.

Combining the estimates obtained above, we deduce for 1 ≤ j < j0

σ2j,t,− ≤ 1√

jexp

(−∣∣v|x|−et√j∣∣2) and σ2

j,t,+ ≤ 1√jexp

(−∣∣v|x|+et√j − 1

∣∣2).When j ≥ j0, we have et|x| ≤

√jv and |x|2 ≤ jv2e−2t < j/4. Therefore

σ2j,t ≤ 2

√j exp(|x|2 − j) < 2

√j exp

(− 3

4j).

Now we are ready to choose the optimal γ’s for the metric integrability condition inthe definition of Markov metric. We respectively define for 1 ≤ j < j0 and j ≥ j0

γ2j,t,ε(x) = exp( |v|x|+ et

√j − 1|2

2

)and γ2j,t(x) =

1√jexp

( j4

).

Then it turns out that

supx∈R

0<t< 136

∑1≤j<j0

σ2j,t,−(x)γ

2j,t,−(x) ≤ sup

x∈R0<t< 1

36

∑1≤j<j0

1√jexp

(− |6v|x| − et

√j|2

4

)

≤ supx∈R

0<t< 136

2

∫Rexp

(− |6v|x| − etu|2

4

)du < ∞.

supx∈R

0<t< 136

∑1≤j<j0

σ2j,t,+(x)γ

2j,t,+(x) ≤ sup

x∈R0<t< 1

36

∑1≤j<j0

1√jexp

(− |v|x|+ et

√j − 1|2

2

)

≤ 1 + supx∈R

0<t< 136

2

∫Rexp

(− |v|x|+ etu|2

2

)du < ∞.

On the other hand, it is clear that

supx∈R

0<t< 136

∑j≥j0

σ2j,t(x)γ

2j,t(x) ≤

∑j≥j0

2 exp(− j2) < ∞.

We have constructed a Markov metric for the Ornstein-Uhlenbeck semigroup.

Let us now verify the analytic conditions, since the algebraic conditions are trivialby commutativity. As we mentioned in Subsection 2.2, the first condition is an easyconsequence of Jensen’s inequality for the Gaussian measure. By the definition ofRj,t,ε and Rj,t, we get qj,t,ε(x, y) = χDε

j,t(x)(y) and qj,t(x, y) = χΣj,t(x)(y). Thus, it

remains to find proper aj,t,ε and aj,t to make sure that

µ(Dεj,t(x)) ≲

∫Ra2j,t,ε(x, y) dµ(y) ≲ γ2j,t,ε(x)µ(D

εj,t(x))


and similarly for the pairs (Σj,t(x)), a2j,t(x, y)). When j < j0 we consider the

functions aj,t,ε(x, y) = χ2Dεj,t(x)

(y), so the lower estimates are trivial. Denote by

cj,t,ε(x) the center of Dεj,t(x). Let β = v(

√j−

√j − 1). Arguing as in (3.6), we get

exp(−∣∣|cj,t,ε(x)|+ β

2

∣∣2) ≤µ(Dε

j,t(x))

β≤ exp

(−∣∣|cj,t,ε(x)| − β

2

∣∣2),exp

(−∣∣|cj,t,ε(x)|+ β

∣∣2) ≤µ(2Dε

j,t(x))

2β≤ exp

(−∣∣|cj,t,ε(x)| − β

∣∣2).Since cj,t,±(x) = et|x| ± 1

2v(√j +

√j − 1), this implies that

µ(2Dεj,t(x))

µ(Dεj,t(x))

≤ 2 exp(3|cj,t,ε(x)|β − 3

4β2)

≤ 2γ2j,t,ε(x).

The estimate for j ≥ j0 is easier. Take aj,t(x, y) = χ2Σj,t(x)(y). Note

2√jv exp

(− |2

√jv|2

)≤ µ(Σj,t(x)) ≤ 2

√jv,

4√jv exp

(− |3

√jv|2

)≤ µ(2Σj,t(x)) ≤ 4

√jv.

Then, since 0 < 2t < 118 , we get

µ(2Σj,t(x))

µ(Σj,t(x))≤ 2 exp(4jv2) ≤ 2 exp(

j

4) = 2γ2j,t(x).

Our choice for qt and at correspond to the balls Σ1,t(x) and 2Σ1,t(x).

The case n > 1. The argument is similar, so we just point out the necessarymodifications. Since v(

√j −

√j − 1) < vj−

12 , we may pick cnj

n−1 balls Dsj,t(x) for

1 ≤ s ≤ cnjn−1 with radius v

2√j, centered on the sphere

y : |y − etx| = v(√j +

√j − 1)

2

such that

Ωj,t(x) ⊂∪s≥1

Dsj,t(x)

and each Ds0j,t(x) overlaps with at most c′n other balls Ds

j,t(x). Then, if f ≥ 0

Otf(x) ≲1

vn

( ∑j<j0

1≤s≤cnjn−1

exp(|x|2 − j)

∫Ds

j,t(x)

fdµ+∑j≥j0

exp(|x|2 − j)

∫Σj,t(x)

fdµ).

Then, we may consider the following Markov metric(σ2j,t,s(x), γ

2j,t,s(x), Rj,t,sf(x)

)=

(exp(|x|2 − j)

vnµ(Ds

j,t(x)), j−n−1

2 exp( |v|x|+ et

√j − 1|2

2

),−∫Ds

j,t(x)

fdµ)

for j < j0 and 1 ≤ s ≤ cnjn−1. When j ≥ j0, we set(

σ2j,t(x), γ

2j,t(x), Rj,t,f(x)

)=(exp(|x|2 − j)

vnµ(Σj,t(x)), j

−n2 exp(

j

4),−∫Σj,t(x)

fdµ).

The analytic conditions hold under the same choices we made for n = 1.


Corollary 3.5. Let O = (Ot)t≥0 be the Ornstein-Uhlenbeck semigroup and T be asingular integral operator defined on L∞(Rn, dµ) with kernel k. More precisely, wehave the kernel representation

Tf(x) =

∫Rn

k(x, y)f(y) dµ(y) for x /∈ suppf.

Suppose T is bounded on L2(Rn, µ) and it satisfies

(3.8) supB ball

supz∈Bj≥1

∫2j+1B\2jB

|k(z, y)|dµ(y) <∞,

(3.9) supB ball

supz1,z2∈B

∫(5B)c

|k(z1, y)− k(z2, y)|dµ(y) <∞.

Then T is a bounded map from L∞(Rn, µ) to the semigroup BMOO space.

Proof. It suffices to prove that our CZ conditions hold. The row and columnboundedness conditions reduce to L2-boundedness. Let Mt be the averaging mapin L∞(Rn, µ) over the ball Σ1,t(x). Given 1 ≤ j < j0, define Aj,t,s(x, ·) as thecharacteristic function over the ball Σj+1,t(x). Then Aj,t,s ≤ χ2rDs

j,t(x)∩Σ1,t(x) with

r = [2 log2(j + 1)] + 3, where [ ] stands for the integer part. Therefore, applying(3.8), we have

supz∈Σ1,t(x)

∫Σj+1,t(x)\2Σ1,t(x)

|k(z, y)| dµ(y) ≲ r ≲ γ2j,t,s(x),

supz∈Ds

j,t(x)

∫Σj+1,t(x)\2Ds

j,t(x)

|k(z, y)| dµ(y) ≲ r ≲ γ2j,t,s(x).

For j ≥ j0, let Aj,t(x, ·) = χΣ4j,t(x) ≤ χ2uΣ1,t(x) with u = [log2(2√j)]+1. Applying

(3.8) as above, we see that T satisfies our size conditions. Moreover, (3.9) impliesour smoothness conditions as in the Euclidean-Lebesguean setting, Section 2.4.

Remark 3.6. Since the Gaussian measure is non-doubling, the term Rj,tf −Mtfin the Markov metric BMO space BMOQ is essential to characterise the changesof the mean values of the function f . This explains the relevance of the size kernelcondition in the Calderon-Zygmund theory for the gaussian measure.

4. Applications II — Noncommutative spaces

In this section we apply our algebraic approach to study Calderon-Zygmundoperators in flag von Neumann algebras which originally motivated us and includematrix algebras, quantum Euclidean spaces and quantum groups. We start byreconstructing and refining the semicommutative theory, which deals with tensorand crossed products with metric measure spaces.

4.1. Operator-valued theory. Let (Ω, µ) be a doubling metric space —as inRemark 1.6— and consider a Markov semigroup St : L∞(Ω) → L∞(Ω). Let M bea semifinite von Neumann algebra with a n.s.f. trace τ . Then we call the semigroup


S = (St ⊗ idM)t≥0 a semicommutative Markov semigroup. Consider the algebra ofessentially bounded functions f : Ω → M equipped with the trace

φ(f) =

∫Ω

τ(f(y)) dµ(y).

Its weak-∗ closure R = L∞(Ω)⊗M is a von Neumann algebra. Assume that thereexists a Markov metric Q = (Rj,t, σj,t, γj,t) : (j, t) ∈ Z+ × R+ associated tothe original Markov semigroup on L∞(Ω). Let qj,t(x, y) = χΩx

j,t(y) stand for the

projections determined by Q via (2.1). We assume in addition that Q satisfies themetric/measure growth condition

(4.1)µ(Σx

j,t)

µ(Ωxj,t)

≤ γj,t(x)

by choosing aj,t(x, y) = χΣxj,t(y). The remaining algebraic and analytic conditions

trivially hold in this case. Indeed, the algebraic conditions follow by commutativityand analytic conditions just require to pick the right averaging maps according toJensen’s inequality, as explained in (2.4). Note that Q satisfies an operator-valuedgeneralization of the Hilbert module majorization in the line of Remark 1.5. ThusQ extends to a Markov metric in R by tensorizing with idM and 1M respectively.

Our goal is to study CZO’s formally given by

Tf(x) =

∫Ω

k(x, y) (f(y)) dµ(y) with

f : Ω → M1 and x /∈ suppΩf,

k(x, y) ∈ L(L0(M1), L0(M2)).

That is, k(x, y) is linear from τ1-measurable to τ2-measurable operators. If weset Rj = L∞(Ω)⊗Mj , we should emphasize that Lp(Rj) = Lp(Ω;Lp(Mj)). Inparticular, this framework does not fall in the vector-valued theory because wetake values in different Banach spaces for different values of p, see [49] for furtherexplanations. This class of operators is inspired by two distinguished examples withM1 = M = M2:

• Operator-valued case

Tf(x) =

∫Ω

kov(x, y) ·f(y) dµ(y).

• Noncommutative model

Tf(x) =

∫Ω

(idM ⊗ τ)[knc(x, y) ·(1M ⊗ f(y))

]dµ(y).

In the first case, the kernel takes values in M or even in the complex field and actson f(y) by left multiplication k(x, y)(f(y)) = kov(x, y) · f(y). It is the canonicalmap when Lp(R) is regarded as the Bochner space Lp(Ω;Lp(M)). On the contraryif we simply think of Lp(R) as a noncommutative Lp space, a natural CZO shouldbe an integral map with respect to the full trace φ =

∫Ω⊗τ and the kernel should

be a φ⊗ φ-measurable operator k : Ω×Ω → M⊗M. The noncommutative modelprovides the resulting integral formula. Note that this model also falls in our generalframework by taking k(x, y)(f(y)) = (idM ⊗ τ)[knc(x, y) · (1M ⊗ f(y))].

Theorem 4.1. Let S = (St)t≥0 be a Markov semigroup on (Ω, µ) which admitsa Markov metric Q = (Rj,t, σj,t, γj,t) : (j, t) ∈ Z+ × R+ satisfying the above


assumptions. Let aj,t(x, y) = χΣxj,t(y)

be the projections determined by Q via (4.1).

Consider the CZO formally given by

Tf(x) =

∫Ω

k(x, y)(f(y)) dµ(y).

Then, T maps L∞(R1) to BMOcS(R2) provided the conditions below hold

i) Lc2-boundedness condition,∥∥∥(∫

Ω

|Tf |2 dµ) 1

2∥∥∥M2

≲∥∥∥(∫

Ω

|f |2 dµ) 1

2∥∥∥M1

.

ii) Smoothness condition for the kernel,∫Ω\Σx

j,t

∥∥(k(y1, z)− k(y2, z))(f(z)

)∥∥M2

dµ(z) ≲ ∥f∥R1

uniformly in j ≥ 1, t > 0, x ∈ Ω and y1, y2 ∈ Ωxj,t.

Proof. The proof follows from Theorem 2.1. Since the underlying space (Ω, µ) is adoubling metric space, the size kernel condition is unnecessary. Thus, it remains tocheck the Lc

2-boundedness condition and the kernel smoothness condition. ConsiderNπ = L∞(Ω× Ω)⊗M1, Nρ = L∞(Ω× Ω)⊗M2, ω(φ)(x, y) = φ(y) for φ ∈ L∞(Ω)

and (π2, ρ2) = (ω ⊗ idM1 , ω ⊗ idM2). Let T = idΩ ⊗ T , Φj,t be the averaging mapover Ωx

j,t × Ωxj,t and ∆ = δ ⊗ idM2 with δφ(x, y) = φ(x)− φ(y). Then condition i)

yields the Lc2-boundedness condition. It is also easy to see that condition ii) implies

our kernel smoothness condition. Thus, the result follows from Theorem 2.1.

Remark 4.2. We continue with a few comments:

A) When M1 = M2 = M and the kernel k(x, y)(f(y)) = k(x, y) · f(y) acts byleft multiplication, the boundedness condition i) becomes equivalent to the usualL2 boundedness. Indeed, using that M ⊂ B(L2(M)) we obtain∥∥∥(∫

Rn

|Tf(y)|2 dy) 1

2∥∥∥M

= sup∥h∥2≤1

(∫Rn

⟨h, |Tf(y)|2h

⟩dy) 1

2

= sup∥h∥≤1

∥∥(Tf) (1Rn ⊗ h)∥∥L2(R)

= sup∥h∥≤1

∥∥T (f (1Rn ⊗ h))∥∥L2(R)

≤ ∥T∥B(L2(R))

∥∥∥(∫Rn

|f(y)|2 dy) 1

2∥∥∥M.

B) We have used so far semigroup type BMO’s. When (Ω, µ) comes equippedwith a doubling metric, we may replace it by other standard (equivalent) formsof BMO, as pointed in Remark 1.7. By well-known arguments [49], our kernelsmoothness condition reduces to

(Smλ) supR>0

ess supy1,y2∈BR

∥∥∥∫(BλR)c

(k(y1, z)− k(y2, z)

)(f(z)) dz

∥∥∥M

≲ ∥f∥R.

for λ > 1. The classical Hormander condition

(Hrλ) ess supy1,y2∈Rn

∫d(y1,z)>λd(y1,y2)

∥∥k(y1, z)− k(y2, z)∥∥M dz <∞.


satisfies (Hrλ) ⇒ (Sm2λ+1). In fact, an even weaker condition suffices

supR>0

∥∥∥−∫BR×BR

∣∣∣ ∫(BλR)c

(k(y1, z)− k(y2, z)

)f(z)dz

∣∣∣2dy1dy2∥∥∥M

≲ ∥f∥2R.

C) We recall that L∞(R) → BMOS boundedness requires that T †f = T (f∗)∗

satisfies the same assumptions as T . If k(x, y) ∈ M is given by left multiplicationthe only effect in T † is that k(x, y) is replaced by k(x, y)∗ and now operates byright multiplication. This left/right condition was formulated in [49] in terms ofM-bimodular maps. Moreover, a counterexample was constructed to show thatthe bimodularity is indeed essential. It is also quite interesting to note that in the‘noncommutative model’ we have∫Rn

(idM ⊗ τ)[k(x, y) ·(1M ⊗ f(y))

]dy =

∫Rn

(idM ⊗ τ)[(1M ⊗ f(y)) · k(x, y)

]dy

by traciality and this pathology does not occur. Finally, the Lp boundedness isguaranteed for 2 < p <∞ since the classical heat semigroup has a regular Markovmetric and Jp = idLp(Rn) in this case. As for 1 < p < 2, it suffices to take adjointswhich leads to Hormander smoothness in the second variable

ess supz1,z2∈Rn

∫|y−z1|>λ|z1−z2|

∥∥k(y, z1)− k(y, z2)∥∥M dy <∞.

Of course, this is still consistent with the classical CZ theory M = C.D) Our analysis of the semicommutative case from our basic Theorem 4.1 does

not recover the weak type (1, 1) inequality from [49]. It requires quasi-orthogonalitymethods which are still missing for general von Neumann algebras.

We now study the L∞ → BMO boundedness of twisted CZO’s on homogeneousspaces. Given a discrete group G with left regular representation λ : G 7→ B(ℓ2(G))let L(G) denote its group von Neumann algebra. Let (M, τ) with M ⊂ B(H) bea noncommutative probability space and α : G 7→ Aut(M) be a trace preservingaction. Consider two ∗-representations

ρ : M ∋ f 7→∑h∈G

αh−1(f)⊗ eh,h ∈ M⊗B(ℓ2(G)),

Λ : G ∋ g 7→∑h∈G

1M ⊗ egh,h ∈ M⊗B(ℓ2(G)),

where eg,h is the matrix unit for B(ℓ2(G)). Now we define the crossed productalgebra M ⋊α G as the weak operator closure in M ⊗ B(ℓ2(G)) of the ∗-algebragenerated by ρ(M) and Λ(G). A generic element of M ⋊α G can be formallywritten as

∑g∈G fg ⋊α λ(g) with fg ∈ M. With this convention, we may embed

the crossed product algebra M ⋊α G into M⊗B(ℓ2(G)) via the map j = ρ ⋊ Λ.Indeed, we have

j(∑

g∈G

fg ⋊α λ(g))

=∑g∈G

ρ(fg)Λ(g)

=∑g∈G

( ∑h,h′∈G

(αh−1(fg)⊗ eh,h)(1M ⊗ egh′,h′))

=∑g∈G

(∑h∈G

αh−1(fg)⊗ eh,g−1h

)


=∑g∈G

(∑h∈G

α(gh)−1(fg)⊗ egh,h

).

Since the action α will be fixed, we relax the terminology and write∑

g∈G fgλ(g)

instead of∑

g∈G fg ⋊α λ(g). We say that a Markov semigroup S = (St)t≥0 in Mis G-equivariant if

αgSt = Stαg for (t, g) ∈ R+ ×G.

If S is a G-equivariant Markov semigroup on M, let S⋊ = (St ⋊ idG)t≥0 andS⊗ = (St ⊗ idB(ℓ2(G)))t≥0 denote the crossed/tensor product amplification of oursemigroup on M ⋊ G and M⊗B(ℓ2(G)) respectively. Note that S⋊ is Markoviandue to the G-equivariance of S. In the following result, our CZO’s are of the form

Tf(x) =

∫Ω

k(x, y)(f(y)) dµ(y)

for all f ∈ (R1, φ1), where (Rj , φj) = L∞(Ω, µ)⊗(Mj , τj) and k(x, y) : M1 → M2.In other words, we keep the same terminology as for Theorem 4.1. We shall alsouse the notation

Mj = Mj⊗B(ℓ2(G)) and Rj = Rj⊗B(ℓ2(G)).

Corollary 4.3. Let G L∞(Ω, µ) be an action α which is implemented by ameasure preserving transformation β, so that αgf(x) = f(βg−1x). Let S = (St)t≥0

be a G-equivariant Markov semigroup on (Ω, µ) which admits a Markov metricQ = (Rj,t, σj,t, γj,t) : (j, t) ∈ Z+ × R+ satisfying the assumptions above. Let usconsider a family of CZO’s formally given by

Tgf(x) =

∫Ω

kg(x, y)(f(y)) dµ(y) for g ∈ G.

Then,∑

g fgλ(g) 7→∑

g Tg(fg)λ(g) is bounded R1 ⋊G → BMOcS⋊

(R2 ⋊G) if

i) Lc2-boundedness condition,∥∥∥(∫

Ω

∣∣(Tgh−1) • ξ∣∣2 dµ) 1

2∥∥∥M2

≲∥∥∥(∫

Ω

|ξ|2 dµ) 1

2∥∥∥M1

,

where • stands for the generalized Schur product of matrices. In otherwords, the CZO Tgh−1 only acts on the (g, h)-th entry of ξ for each g, h ∈ G.

ii) Smoothness condition for the kernel,∫Ω

∥∥(K(y1, z)−K(y2, z))•(ξ(z)(1− aj,t(x, z))

)∥∥M2

dµ(z) ≲ ∥ξ∥R1,

uniformly on j ≥ 1, t > 0, x ∈ Ω and y1, y2 ∈ Ωxj,t. Here, the CZ kernel

K(y, z) =∑

g,h kgh−1(βgy, βgz)⊗eg,h acts once more as a Schur multiplier.

Proof. Letting ξ =∑

g,h ag,h ⊗ eg,h ∈ R1, we define the map

Φ : R1 → BMOcS⊗

(R2),

Φ(ξ)(x) =∑g,h

αg−1

∫Ω

kgh−1(x, y)(ag,h(βg−1(y)))dµ(y)⊗ eg,h.


By the definition of j, it is easy to check that

j(∑

g

Tg(fg)λ(g))

= Φ(j(∑

g

fgλ(g))).

Since S is G-equivariant, according to [33, Lemma 2.1], we have

∥g∥BMOcS⋊(R2⋊G) = sup

t≥0

∥∥∥(S⊗,t|j(g)|2 − |S⊗,tj(g)|2) 1

2∥∥∥R2

.

Therefore, it suffices to show that Φ is R1 → BMOcS⊗

(R2) bounded. We find

Φ(ξ)(x) =

∫Ω

K(x, y)(ξ(y))dµ(y).

Thus, we may regard Φ as a semicommutative CZO and apply Theorem 4.1 where

Mj is replaced by Mj . Since Φ(ξ) =∑

g,h(αg−1)•(Tgh−1)•(αg)•ξ and β is measurepreserving, we immediately find that the Lc

2-boundedness assumption implies thatthe map

Φ : Lc2(Ω)⊗M1 → Lc

2(Ω)⊗M2

is bounded. Moreover, the smoothness condition matches that of Theorem 4.1. Remark 4.4. Our work so far yields sufficient conditions for the L∞ → BMOboundedness of T ⋊ idG in more general settings. In particular, if Tg = T andαgT = Tαg for all g ∈ G, then we find for any T fulfilling the assumption ofTheorem 4.1, T ⋊ idG : R1 ⋊G 7→ BMOc

S⋊(R2 ⋊G) is bounded.

4.2. Matrix algebras. In this paragraph, we introduce a Markov metric for thematrix algebra B(ℓ2). The triangular truncation plays the noncommutative formof the Hilbert transform on B(ℓ2). We shall reprove the Lp-boundedness of thetriangular truncation for 1 < p < ∞ and a new BMO → BMO estimate by meansof this Markov metric and our algebraic approach. Consider the ∗-homomorphismu : B(ℓ2) → L∞(R)⊗B(ℓ2) determined by

u(emk) = e2πi(m−k) ·emk.

Given A =∑

m,k amkemk, define the semigroup

St(A) =∑

m,ke−t|m−k|2amkemk.

It is not difficult to see that it defines a Markov semigroup of convolution type. Infact, u is a corepresentation of L∞(R) (equipped with its natural comultiplicationmap ∆f(x, y) = f(x + y)) in B(ℓ2) and it turns out that S = (St)t≥0 is thetransferred semigroup associated to the heat semigroup on R

u St = (Ht ⊗ idB(ℓ2)) u.

Define the cpu map Rj,t on B(ℓ2) by u Rj,t = (Rj,t ⊗ idB(ℓ2)) u, where Rj,tf(x)denotes the average of f ∈ L∞(R) over the interval B√

4jt(x). Now, given a matrixA =

∑m,k amkemk we find

u Rj,t(A)(x) = −∫B√

4jt(x)

u(A)(y)dy

= −∫B√

4jt(x)

∑m,k

e2πi(m−k)yamkemkdy


=∑m,k

sin(4√jtπ(m− k))

4√jtπ(m− k)

e2πi(m−k)xamkemk.

Thus, we find the following identity

Rj,t(A) =∑m,k

sin(4√jtπ(m− k))

4√jtπ(m− k)

amkemk.

Taking σ2j,t = 2e

√j/πe−j1B(ℓ2) and γ

2j,t =

√j1B(ℓ2), we obtain a Markov metric in

B(ℓ2). Indeed, the metric integrability condition holds trivially, as for the Hilbertmodule majorization it reduces to prove that B1 ≤ B2 with

B1 = u(⟨ξ, ξ⟩St

)=⟨u⊗ u(ξ), u⊗ u(ξ)

⟩Ht⊗idB(ℓ2)

,

B2 =∑j

σ2j,tu(⟨ξ, ξ⟩Rj,t

)=∑j

σ2j,t

⟨u⊗ u(ξ), u⊗ u(ξ)

⟩Rj,t⊗idB(ℓ2)

.

In other words, it suffices to note that the canonical Markov metric in R —whichrecovers the Euclidean metric, as proved in Paragraph 1.4— admits a matrix-valuedextension, as it was justified in Remark 1.5. Let us now consider the triangulartruncation

(A) =∑m>k

amkemk.

Corollary 4.5. We have

∥(A)∥BMOS ≲ ∥A∥BMOS .

In particular, given 1 < p <∞ we obtain ∥(A)∥Sp≲ p2

p− 1∥A∥Sp

.

Proof. Recall thatu = (L⊗ idB(ℓ2)) u.

for L = 12 (id + iH) and Hf(ξ) = −isgn(ξ)f(ξ), the Hilbert transform in the real

line. We may also regard u : B(ℓ2) → L∞(T)⊗B(ℓ2) as a corepresentation of Tinstead of R and the above identity holds replacing H by the Hilbert transformin the torus. In this case, u becomes a trace preserving ∗-homomorphism andthe well-known Sp inequalities for reduce to the boundedness of the Hilberttransform in Lp(T;Sp(ℓ2)), which is also well-known and follows in passing fromthe semicommutative theory in the previous paragraph. Alternatively, the secondassertion follows from the first one by interpolation and duality. According toRemark 1.7, to prove the first assertion it suffices to show that the map

T = i(idB(ℓ2) − 2)

is BMO → BMO bounded for the semigroup BMO space which is associated to thetransferred Poisson semigroup Pt on B(ℓ2) given by

Pt : (aij) 7→ (e−t|i−j|aij).

Given A = (ajk)j,k in B(ℓ2) then

|A|2 = A∗A =(∑

k

akiakj

)i,j

T (A) = i(sgn(k − j)ajk

)j,k

(TA)∗ = i(sgn(k − j)akj

)j,k


Then(Pt|A|2 − |PtA|2

)ij=∑

k(e−t|i−j| − e−t|k−j|e−t|i−k|)akiakj and(

Pt|T (A)|2 − |PtT (A)|2)ij

=∑k

(e−t|i−j| − e−t|k−j| e−t|i−k|) sgn(k − i) sgn(k − j) akiakj .

Since sgn(k − i)sgn(k − j) = 1 iff e−t|i−k|e−t|k−j| = e−t|i−j|, we get

Pt|A|2 − |PtA|2 = Pt|T (A)|2 − |PtT (A)|2.

The last identity implies that T is an isometry on the Poisson BMO space.

4.3. Quantum Euclidean spaces. Given an integer n ≥ 1, fix an anti-symmetricR-valued n× n matrix Θ. We define AΘ as the universal C*-algebra generated bya family u1(s), u2(s), · · · , un(s) of one-parameter unitary groups in s ∈ Rn whichare strongly continuous and satisfy the following Θ-commutation relations

uj(s)uk(t) = e2πiΘjkstuk(t)uj(s).

If Θ = 0, by Stone’s theorem we can take uj(s) = exp(2πis⟨ej , ·⟩) and AΘ is thespace of bounded continuous functions on Rn. In general, given ξ ∈ Rn, definethe unitaries λΘ(ξ) = u1(ξ1)u2(ξ2) · · ·un(ξn). Let EΘ be the closure in AΘ ofλΘ(L1(Rn)) with

f =

∫Rn

fΘ(ξ)λΘ(ξ) dξ.

If Θ = 0, EΘ = C0(Rn). Define

τΘ(f) = τΘ

(∫Rn

fΘ(ξ)λΘ(ξ) dξ

)= fΘ(0)

for fΘ : Rn → C integrable and smooth. τΘ extends to a normal faithful semifinitetrace on EΘ. Let RΘ = A′′

Θ = E′′Θ be the von Neumann algebra generated by EΘ

in the GNS representation of τΘ. Note that if Θ = 0, RΘ = L∞(Rn). In generalwe call RΘ a quantum Euclidean space. There are two maps which play importantroles while doing analysis over quantum Euclidean spaces. The first one is thecorepresentation map σΘ : RΘ → L∞(Rn)⊗RΘ, given by λΘ(ξ) 7→ expξ ⊗λΘ(ξ)where expξ stands for the Fourier character exp(2πi⟨ξ, ·⟩). Note that σΘ is a normalinjective ∗-homomorphism. The second map is πΘ : expξ 7→ λΘ(ξ)⊗ λΘ(ξ)

∗, which

extends to a normal ∗-homomorphism from L∞(Rn) to RΘ⊗RopΘ , where Rop

Θ isthe apposite algebra of RΘ, which is obtained by preserving the linear and adjointstructures but reversing the product. We refer the readers to [24] for more detailedinformation of quantum Euclidean spaces and these two maps.

BMO and Markov metric. Our first goal is to construct a natural Markov metricfor quantum Euclidean spaces. Let us recall the heat semigroup on Rn acting onφ : Rn → C admits the following form

Htφ(x) =

∫Rn

φ(ξ)e−t|ξ|2 expξ(x) dξ.

This induces a semigroup on RΘ determined by

σΘ SΘ,t = (Ht ⊗ idRΘ) σΘ.


SΘ,t gives a Markov semigroup on RΘ which formally acts as

(4.2) SΘ,t(f) =

∫Rn

fΘ(ξ)e−t|ξ|2λΘ(ξ) dξ.

The corresponding semigroup column BMO norm is given by

∥f∥BMOc(RΘ) = supt>0

∥∥∥(SΘ,t(|f |2)− |SΘ,t(f)|2) 1

2∥∥∥RΘ

≈ supBball inRn

∥∥∥(−∫B

|σΘ(f)− σΘ(f)B|2dµ) 1

2∥∥∥RΘ

= ∥σΘ(f)∥BMOc(Rn;RΘ).

According to Remark 1.5, the semicommutative extension Ht ⊗ idRΘof the heat

semigroup, together with the extension of the corresponding Markov metric fromParagraph 2.4 still satisfies the Hilbert module majorization

(4.3) ⟨ξ, ξ⟩Ht⊗idRΘ≤∑j≥1

σ∗j,t⟨ξ, ξ⟩Rj,t⊗idRΘ

σj,t

as well as the integrability condition, where σ2j,t ≡ 2e

√jn/πe−j , γ2j,t ≡ j

n2 and

Rj,tf(x) is the average of f over B√4jt(x). Then we can easily produce a Markov

metric on RΘ. Let Bj,t be the Euclidean ball in Rn centered at the origin withradius

√4jt and consider the projections qj,t = χBj,t

⊗ 1RΘ. Define the cpu maps

RΘ,j,t(f) =1

|Bj,t|

∫Bj,t

σΘ(f)(x) dx =1

|Bj,t|

∫Rn

χBj,t(ξ)fΘ(ξ)λΘ(ξ) dξ.

It is easy to check that

(4.4) σΘ RΘ,j,t = (Rj,t ⊗ idRΘ) σΘ.

The Hilbert module majorization

⟨ξ, ξ⟩SΘ,t≤∑j≥1

σ∗j,t⟨ξ, ξ⟩RΘ,j,t

σj,t

for ξ ∈ RΘ⊗SΘ,tRΘ is equivalent to the same inequality after composing with the∗-homomorphism σΘ, which follows in turn by the intertwining identities (4.2) and(4.4), together with the majorization (4.3). Therefore, we obtain a Markov metricon RΘ associated to SΘ

QΘ =(RΘ,j,t, σj,t, γj,t) | (j, t) ∈ Z+ × R+

.

The algebraic structure. We start with the kernel representation of our CZOsover the (fully noncommutative) von Neumann algebra RΘ. Given a kernel kaffiliated to RΘ⊗Rop

Θ , the linear map associated to it is formally given by

Tkf = (idRΘ⊗ τΘ)

(k(1RΘ

⊗ f))= (idRΘ

⊗ τΘ)((1RΘ

⊗ f)k).

The reader is referred to [24] for more details. Our goal is to provide sufficientconditions for the L∞ → BMO boundedness of Tk. Consider the ∗-homomorphismσΘ : RΘ → L∞(Rn)⊗RΘ. In the case of quantum Euclidean spaces, we need thefull algebraic skeleton introduced in Section 2. In Table 1 there is a little dictionaryto identify the main objects. Next, note that

σΘ Tk(f) = (idRn ⊗ idRΘ⊗ τΘ)

(kσ(1Rn ⊗ 1RΘ

⊗ f)),


Generic algebraic objects Quantum Euclidean spaces

M RΘ

Nρ

L∞(Rn)⊗RΘ

ρ1 = 1⊗ ·, ρ2 = σΘEρ = Lebesgue integral

Nπ

RΘ⊗RopΘ

π1 = 1⊗ ·, π2 = · ⊗ 1Eπ = τΘ ⊗ idRop

Θ

Nσ

L∞(Rn)⊗L∞(Rn)⊗RΘ

Nσ = (δ ⊗ idRΘ)(Nρ)δφ(x, y) = φ(x)− φ(y)

Table 1. Algebraic skeleton for RΘ.

where kσ = (σΘ ⊗ idRopΘ)(k). Denote σΘ Tk by Tkσ

. Define

Tk : RΘ⊗RΘ ∋ f ⊗ a 7→ Tkσ(f)(1Rn ⊗ a) ∈ L∞(Rn)⊗RΘ.

Then it is clear that the compatibility condition (2.3) holds since Tk π2 = σΘ Tk.

Lemma 4.6. If Tk is bounded on L2(RΘ), then∥∥Tk : Lc2(RΘ)⊗RΘ → Lc

2(Rn)⊗RΘ

∥∥ ≤∥∥Tk : L2(RΘ) → L2(RΘ)

∥∥.Proof. We need to introduce two maps:

jΘ : L2(Rn) ∋∫Rn

φ(ξ) expξ dξ 7→∫Rn

φ(ξ)λΘ(ξ) dξ ∈ L2(RΘ),

W : Lc2(Rn)⊗RΘ ∋

∫Rn

expξ ⊗a(ξ) dξ 7→∫Rn

expξ ⊗λΘ(ξ)a(ξ) dξ ∈ Lc2(Rn)⊗RΘ.

It is straightforward to show that W extends to an isometry. Moreover, jΘ is alsoan L2-isometry, we refer the reader to [24, Section 1.3.2] for the proof. Observethat

σΘ(f)(1Rn ⊗ a) =

∫Rn

fΘ(ξ) expξ ⊗λΘ(ξ)a dξ

= W (

∫Rn

fΘ(ξ) expξ ⊗ a dξ) = W (j∗Θ ⊗ idRΘ)(f ⊗ a).

Letting f = Tkg, we get

Tk(g ⊗ a) =W (j∗ΘTk ⊗ idRΘ)(g ⊗ a).

The properties of the maps jΘ and W readily imply the assertion.

Now let us introduce a weak-∗ dense subalgebra of RΘ, which is the analogue ofthe classical Schwartz class. Let S(Rn) denote the classical Schwartz class in theEuclidean space Rn and define

SΘ =f ∈ RΘ : fΘ ∈ S(Rn)

.


Let S ′Θ denote the space of continuous linear functionals on SΘ, which is the

quantum space of tempered distributions. Consider a continuous linear operatorT ∈ L(SΘ,S ′

Θ). By using the unitary map

jΘ : S(Rn) → SΘ

as defined in the proof of Lemma 4.6, we get j∗ΘTjΘ ∈ L(S(Rn),S(Rn)′). By aresult of Schwartz, there exists a unique kernel K ∈ S ′(R2n) = (S(Rn)⊗π S(Rn))′

such that T admits the kernel k = (jΘ ⊗ jΘ)(K) ∈ (SΘ ⊗π SΘ)′. Actually, the

kernel representations Tk satisfying the Calderon-Zygmund type conditions in thefollowing theorem belong to L(SΘ,S ′

Θ). It provides sufficient conditions for theL∞(RΘ) → BMOc(RΘ) boundedness of CZO operators associated to kernels in(SΘ⊗π SΘ)

′. We shall use the quantum analogue of the bands around the diagonal

aB = πΘ(χ5B) =

∫Rχ5B(ξ)λΘ(ξ)⊗ λΘ(ξ)

∗ dξ.

Theorem 4.7. Let Tk ∈ L(SΘ,S ′Θ) and assume

i) Cancellation ∥∥Tk : L2(RΘ) → L2(RΘ)∥∥ <∞.

ii) For any f ∈ RΘ and any Euclidean ball B centered at the origin

−∫B×B

∣∣ΣΘ,k,f,B(y1)− ΣΘ,k,f,B(y2)∣∣2dy1dy2 ≲ ∥f∥2RΘ

,

where ΣΘ,k,f,B = (idRn ⊗ idRΘ⊗ τΘ)

[kσ(1Rn ⊗ 1RΘ

⊗ f)(1Rn ⊗ a⊥B)].

Then, the Calderon-Zygmund operator Tk is bounded from L∞(RΘ) to BMOc(RΘ).

Proof. By Theorem 1.4, it suffices to prove

Tk : L∞(RΘ) → BMOcQΘ.

Arguing as in Paragraph 1.4, the Markov metric BMO norm takes the simpler form

∥f∥BMOcQΘ

= supt>0

supj≥1

∥∥∥(γ−1j,t

[RΘ,j,t(|f |2)− |RΘ,j,t(f)|2

]γ−1j,t

) 12∥∥∥RΘ

.

In other words, the extra term in the definition of BMO is dominated by theabove expression as in (1.3). As noticed in Remark 2.5, the size kernel condition isthen superfluous. This also reduces the analytic conditions and the smooth kernelconditions to be checked. In summary, according to the proof of Theorem 2.1, theassertion will follow if we can justify:

C0) Initial condition

Tk : AΘ → RΘ for AΘ ⊂ RΘ weak-∗ dense.

Al1) QΘ-monotonicity of Eρ

Eρ(qj,t|ξ|2qj,t) ≤ Eρ(|ξ|2).

Al2) Right modularity of Tk

Tk(ηπ1(b)) = Tk(η)ρ1(b).


An1) Mean differences

RΘ,j,t(ξ∗ξ)− RΘ,j,t(ξ)

∗RΘ,j,t(ξ) ≤ Φj,t

(δ(ξ)∗δ(ξ))

)for some cpu Φj,t.

An2) Metric/measure growth

1 ≤ π1ρ−11 Eρ(qj,t)

− 12Eπ(a

∗j,taj,t)π1ρ

−11 Eρ(qj,t)

− 12 ≲ π1ρ

−11 (γ2j,t).

CZ1) Lc2-boundedness condition

Tk : Lc∞(Nπ;Eπ) → Lc

∞(Nρ;Eρ).

CZ2) Kernel smoothness condition

Φj,t

(∣∣δ(Tk(π2(f)(1 − aj,t)))∣∣2) ≲ γ2j,t∥f∥2∞.

The initial condition trivially holds for good kernels k ∈ SΘ ⊗alg SΘ. In [24] itwas required to extend the main result from this class of kernels to general onesin S ′

Θ⊕Θ, by reproving certain auxiliary results in the context of distributions. Inour case, this is much simpler. Indeed, when dealing with general kernels, we justnote that Tk(f) ∈ L2(RΘ) for all f ∈ SΘ by assumption. Given the form of RΘ,j,t,it trivially follows that RΘ,j,t(|Tkf |2) and RΘ,j,tTkf are well-defined operators inL1(RΘ) and L2(RΘ) respectively. In particular, the proof of Theorem 2.1 followsexactly as it was written there under this more flexible assumption. Therefore, theinitial condition can be relaxed to the condition

Tk : SΘ → L2(RΘ).

In fact, according to [24, Proposition 2.17], every algebraic column CZO is normal.Thus, it suffices —as we did in Theorem 2.1— to justify that Tk : SΘ → BMOc

QΘ

is bounded, as we shall do by justifying the remaining conditions.

Al1 holds trivially since qj,t = χBj,t⊗1RΘ

lives in the center of Nρ. On the otherhand, according to the definition of ρ1, π1 from Table 1, the algebraic condition Al2can be rewritten as follows

Tk(η(1RΘ ⊗ b)

)= Tk(η)(1Rn ⊗ b).

This is clear from the definition of Tk. Next, condition An1 reads as

−∫Bj,t

|ξ|2dµ−∣∣∣−∫

Bj,t

ξdµ∣∣∣2 ≤ −

∫Bj,t×Bj,t

∣∣ξ(x)− ξ(y)∣∣2dµ(x)dµ(y)

for RΘ-valued functions, when Φj,t is chosen to be the average over Bj,t × Bj,t.As in (2.4), this is a consequence of the operator-valued Jensen’s inequality. Nextrecalling that aj,t = πΘ(χ5Bj,t), condition An2 takes the form

|Bj,t|1RΘ ≤ (τΘ ⊗ idRopΘ)(πΘ(χ5Bj,t)) ≲ j

n2 |Bj,t|1RΘ

.

To verify it we note that

(τΘ ⊗ idRopΘ)(πΘ(φ)) = (τΘ ⊗ idRop

Θ)(∫

Rn

φ(ξ)λΘ(ξ)⊗ λΘ(ξ)∗dξ)= φ(0)1RΘ

.

Then we get (τΘ ⊗ idRopΘ)(πΘ(χ5Bj,t

)) = 5|Bj,t|1RΘ. Condition CZ1 reduces to our

L2-boundedness assumption by Lemma 4.6. Finally, the smoothness condition ii)in the statement readily implies condition CZ2 for all values of j, t.


The smoothness condition in Theorem 4.7 is of Hormander type, while the onein the main result of [24] is a gradient condition. As expected, we shall show thatour condition in this paper is more flexible than that of [24, Theorem 2.6]. We use• for the product in M⊗Mop, so that

(a⊗ b) • (a′ ⊗ b′) = (aa′)⊗ (b′b).

The quantum analogue of the metric is defined by

dΘ = πΘ(| · |)

for the Euclidean norm | · |. Moreover, we also introduce the Θ-deformation of thefree gradient. Let L(Fn) denote the group von Neumann algebra associated to thefree group with n generators Fn. It is well-known from (say) [64] that L(Fn) isgenerated by n semicircular random variables s1, s2, . . . , sn. Note that there existderivations ∂jΘ in SΘ which are determined by

∂jΘ(λΘ(ξ)) = 2πiξjλΘ(ξ)

for 1 ≤ j ≤ n. Define the Θ-deformed free gradient as

∇Θ =

n∑j=1

sj ⊗ ∂jΘ : SΘ → L(Fn)⊗RΘ.

If ∇ denotes the free gradient for Θ = 0, it is easy to check that

(idL(Fn) ⊗ σΘ) ∇Θ =

n∑j=1

sj ⊗ (σΘ ∂jΘ)(4.5)

=

n∑j=1

sj ⊗ (∂j σΘ) = (∇⊗ idRΘ) σΘ.

For the convenience of the reader, we cite Theorem 2.6 from [24] below.

Theorem 4.8. Let Tk ∈ L(SΘ,S ′Θ) and assume:

i) Cancellation

∥Tk : L2(RΘ) → L2(RΘ)∥ ≤ A1.

ii) Gradient condition. There exists

α <n

2< β <

n

2+ 1

satisfying the gradient conditions below for ρ = α, β∣∣∣dρΘ • (∇Θ ⊗ idRopΘ)(k) • dn+1−ρ

Θ

∣∣∣ ≤ A2.

Then, we find the following L∞ → BMOc estimate∥∥Tk : L∞(RΘ) → BMOc(RΘ)∥∥ ≤ Cn(α, β)(A1 +A2).

To simplify notation, we shall write in what follows Σ for ΣΘ,k,f,B. According tothe semicommutative Poincare type inequality introduced in [24, Proposition 1.6]we obtain∥∥∥−∫

B×B

∣∣δ(Σ)∣∣2dµ× µ∥∥∥RΘ

≤ 16R2∥∥∥(1⊗ χB ⊗ 1)(∇⊗ idRΘ)(Σ)

∥∥∥2L(Fn)⊗L∞(Rn)⊗RΘ


for R = radius of B. By (4.5), we may rewrite

(1⊗ χB ⊗ 1)(∇⊗ idRΘ)(Σ)

= (id⊗3 ⊗ τΘ)((1⊗ χB ⊗ 1⊗2)(∇⊗ id⊗2)(kσ)(1

⊗3 ⊗ f)(1⊗2 ⊗ a⊥j,t))

= (id⊗3 ⊗ τΘ)((1⊗ χB ⊗ 1⊗2)(id⊗ σΘ ⊗ id)(∇Θ ⊗ id)(k)(1⊗3 ⊗ f)(1⊗2 ⊗ a⊥j,t)

)= (id⊗3 ⊗ τΘ)

(K • (1⊗3 ⊗ f)

)with

K = (1⊗ χB ⊗ 1⊗2)(id⊗ σΘ ⊗ id)(∇Θ ⊗ id)(k) • (1⊗2 ⊗ a⊥j,t)

in L(Fn)⊗(S(Rn) ⊗π SΘ ⊗π SΘ)′. Thus, (1 ⊗ χB ⊗ 1)(∇ ⊗ idRΘ

)(Σ) = TK(f).We turn to the proofs of Theorem 2.6, Proposition 2.15 and Remark 2.16 (as thegeneralizations of Theorem 2.6) in [24], they show that the condition ii) in Theorem4.8 implies ∥∥TK(f)

∥∥L(Fn)⊗L∞(Rn)⊗RΘ

≤ Cn(α, β)A2

R∥f∥RΘ

,

which is inequality (2.2) in [24]. Combining the calculations above, we deducethat condition ii) in Theorem 4.8 is stronger than condition ii) in Theorem 4.7. Inconclusion, the Calderon-Zygmund extrapolation onRΘ that we obtain by applyingTheorem 2.1 improves the corresponding result in [24].

4.4. Quantum Fourier multipliers. We now refine our abstract result for locallycompact quantum groups. We shall need some basic notions from the theory ofquantum groups, details can be found in Kustermans/Vaes’ papers [38, 39]. Letus consider a von Neumann algebra N equipped with a comultiplication map, anormal injective unital ∗-morphism ∆ : N → N⊗N satisfying the coassociativitylaw

(idN ⊗∆)∆ = (∆⊗ idN )∆.

Assume also the existence of two n.s.f weights ψ and φ on N such that

(idN ⊗ ψ)∆(a) = ψ(a)1N and (φ⊗ idN )∆(a) = φ(a)1N for a ∈ N+.

We call ψ and φ the left-invariant Haar weight and the right-invariant Haar weighton N respectively. Then the quadruple G = (N ,∆, ψ, φ) is called a (von Neumannalgebraic) locally compact quantum group and we write L∞(G) for the quantumgroup von Neumann algebra N . Using the Haar weights, one can construct anantipode S on N which is a densely defined anti-automorphism on N satisfying theidentity

(idN ⊗ ψ)((1N ⊗ a∗)∆(b)

)= S

((idN ⊗ ψ)

(∆(a∗)(1N ⊗ b)

)).

The comultiplication map ∆ determines a multiplication on the predual L1(G)given by convolution φ1 ⋆ φ2(a) = (φ1 ⊗ φ2)∆(a). The pair (L1(G), ⋆) forms aBanach algebra. In what follows, if not specified otherwise, the quantum groupsG we shall work with admit a tracial left-invariant Haar weight ψ. The simplestmodel of noncommutative quantum groups are group von Neumann algebras L(G)associated to discrete groups. If λ is the left regular representation of G, thecomultiplication is determined by ∆(λ(g)) = λ(g) ⊗ λ(g). Its isometric naturefollows from Fell’s absorption principle and the convolution is abelian. The standardtrace on L(G) is a left and right-invariant Haar weight. Moreover, in this case, theantipode is bounded and S(λ(g)) = λ(g−1).


A convolution semigroup of states is a family (ϕt)t≥0 of normal states on L∞(G)such that ϕt1 ⋆ ϕt2 = ϕt1+t2 . The corresponding semigroup of completely positivemaps is given by

S∆,t(a) = (ϕt ⊗ idG) ∆(a).

When S∆ = (S∆,t)t≥0 is a Markov semigroup, we call it a convolution semigroup.

Lemma 4.9. Let G be a locally compact quantum group equipped with a convolutionsemigroup of states (ϕt)t≥0. Then, S∆ = ((ϕt⊗ idG)∆)t≥0 is a Markov semigroupon L∞(G) whenever

i) ϕt S = ϕt for all t ≥ 0,ii) S∆,t(a) → a as t→ 0+ in the weak-∗ topology of L∞(G).

Proof. Let us begin with the self-adjointness

ψ(a∗S∆,t(b)

)= ψ

(a∗(ϕt ⊗ idG)∆(b)

)= ϕt ⊗ ψ

((1G ⊗ a∗)∆(b)

)= ϕt

((idG ⊗ ψ)

((1G ⊗ a∗)∆(b)

)= ϕt

(S (idG ⊗ ψ)

(∆(a∗)(1G ⊗ b)

)︸︷︷︸ρ

).

This means that ψ(a∗S∆,t(b)

)= ϕt(S(ρ)) = ϕt(ρ) and we get

ψ(a∗S∆,t(b)

)= ϕt ⊗ ψ

(∆(a∗)(1G ⊗ b)

)= ψ

((ϕt ⊗ idG) ∆(a∗)b

)= ψ

(S∆,t(a)

∗b).

The remainder properties are straightforward. Indeed, identity S∆,t(1G) = 1G isobvious. The weak-∗ convergence of the S∆,t(a)’s as t → 0+ is assumed and thecomplete positivity is clear. The normality follows from the weak-∗ continuity ofϕt and ∆. Finally, the semigroup law easily follows from coassociativity.

In what follows, we shall assume that the hypotheses of Lemma 4.9 hold. Letus fix a quantum group G = (N ,∆, ψ, φ) and consider a convolution semigroup S∆

associated to it. A Markov metric

Q =(Rj,t, σj,t, γj,t) : j, t ∈ Z+ × R+

in L∞(G) = N associated to S∆ will be called an intrinsic Markov metric whenthere exists an increasing family of projections pj,t in L∞(G) such that the cpumaps take the form

(4.6) Rj,tf =1

ψ(pj,t)(ψ ⊗ idG)

((pj,t ⊗ 1G)∆(f)

).

In other words, we use the algebraic skeleton(Nρ = Nπ, ρ1, ρ2,Eρ, qj,t

)=(L∞(G)⊗L∞(G),1⊗ ·,∆, ψ ⊗ idG, pj,t ⊗ 1G

).

Remark 4.10. Assume that

γj,t ∈ R+ and γ2j,t ≥ψ(pj,t)

ψ(p1,t).


Then, the term |Rj,tf −Mtf | in the metric BMO norm satisfies for Mt = R1,t that∣∣Rj,tf −Mtf∣∣2 =

∣∣∣ 1

ψ(p1,t)(ψ ⊗ idG)

((∆(f)− 1⊗Rj,tf)(p1,t ⊗ 1)

)∣∣∣2≤ 1

ψ(p1,t)(ψ ⊗ idG)

[∣∣(∆(f)− 1⊗Rj,tf)(p1,t ⊗ 1)∣∣2]

=ψ(pj,t)

ψ(p1,t)

(Rj,t|f |2 − |Rj,tf |2

)≤ γj,t

(Rj,t|f |2 − |Rj,tf |2

)γj,t.

According to Theorem 1.4, this yields

(4.7) ∥f∥BMOcS∆

≲ ∥f∥BMOcQ

≲ supt>0

supj≥1

∥∥∥(Rj,t|f |2 − |Rj,tf |2) 1

2∥∥∥L∞(G)

.

Additionally, we may consider transferred Markov metrics in other von Neumannalgebras. Consider a convolution semigroup of states (ϕt)t≥0 on a locally compactquantum group L∞(G). A corepresentation π : M → L∞(G)⊗M is a normalinjective ∗-representation satisfying the identity

(idG ⊗ π) π = (∆⊗ idM) π.

Every such π yields a transferred convolution semigroup Sπ = (Sπ,t)t≥0 with

Sπ,t : M → M,

Sπ,tf = (ϕt ⊗ idM) π(f).

Lemma 4.11. Assume that

• τ(Sπ,t(f1)∗f2) = τ(f∗1Sπ,t(f2)),

• Sπ,tf → f as t→ 0+ in the weak-∗ topology of M.

Then Sπ defines a Markov semigroup on M such that π Sπ,t = (S∆,t ⊗ idM) π.

Proof. It is easy to check that Sπ,t is cpu and the normality follows from the weak-∗continuity of ϕt and π. Hence, it remains to show the identity πSπ,t = (St⊗idG)πand the semigroup law. We first observe that π(ϕt⊗idM) = (ϕt⊗idG⊗idM)(idG⊗π)as maps on L∞(G)⊗M. Indeed, by weak-∗ continuity, it suffices to test the identityon elementary tensors n⊗m, for which the identity is trivial. Therefore, we have

(S∆,t ⊗ idM)π = (ϕt ⊗ idG ⊗ idM)(∆⊗ idM)π

= (ϕt ⊗ idG ⊗ idM)(idG ⊗ π)π

= π(ϕt ⊗ idM)π = πSπ,t.

For the semigroup law we note that

Sπ,t1Sπ,t2 = (ϕt1 ⊗ idM)(ϕt2 ⊗ idG ⊗ idM)(idG ⊗ π)π

= (ϕt1 ⊗ idM)(ϕt2 ⊗ idG ⊗ idM)(∆⊗ idM)π

= (ϕt2 ⊗ ϕt1 ⊗ idM)(∆⊗ idM)π = (ϕt2 ⋆ ϕt1 ⊗ idM)π = Sπ,t1+t2 .

In the sequel, we shall assume that the assumptions in Lemma 4.11 hold. IntrinsicMarkov metrics on L∞(G) yield transferred Markov metrics on M associated to thetransferred convolution semigroup Sπ. Indeed, given any intrinsic Markov metric


Q = (Rj,t, σj,t, γj,t) in G with cpu maps Rj,t given by (4.6), the transferred cpumaps Rπ,j,t are given by

Rπ,j,tf =1

ψ(pj,t)(ψ ⊗ idM)

((pj,t ⊗ 1)π(f)

).

It is easy to check that π Rπ,j,t = (Rj,t ⊗ idM) π. Assume in addition thatσj,t ∈ R+. Then, arguing as we did before Corollary 4.5 for the corepresentation uof R in B(ℓ2), we get a Markov metric in M

Qπ =(Rπ,j,t, σj,t, γj,t) : j ∈ Z+, t ∈ R+

.

Let α : N → N be a strictly increasing function with α(j) > j. This Markov metricis called α-doubling if there exists some constant cα such that ψ(qα(j),t) ≤ cαψ(qj,t).

Remark 4.12. In what follows, we impose our Markov metrics to be α-doublingfor some function α : N → N, to satisfy σj,t ∈ R+ as well as the condition inRemark 4.10. Altogether, this allows to eliminate the size CZ condition and reducethe number of analytic and smoothness CZ conditions to be checked for both theintrinsic Markov metric and the transferred one.

Observe that the transferred formulation above includes the intrinsic formulationby taking (M, π) = (G,∆). Let us now state the corresponding Calderon-Zygmundtheory. Given AM a weakly dense ∗-subalgebra of M, let T be a (not necessarilybounded) operator T : AM → M. We say T is a transferred map if there exists anamplification map

T : D ⊂ L∞(G)⊗M → L∞(G)⊗Msatisfying the identity

(4.8) π T = T π|AM.

Again, D is a weakly dense ∗-subalgebra for which π(AM) ⊂ D. In the case(M, π) = (G,∆), we can always take the amplification T⊗idM and condition abovejust imposes that T is a quantum Fourier multiplier. In the following theorem, weprovide sufficient conditions on the amplification map to make a given transferredCZO T bounded from AM to BMOc

Sπ.

Theorem 4.13. Letπ : M → L∞(G)⊗M

be a corepresentation of a locally compact quantum group G in a semifinite vonNeumann algebra (M, τ). Assume that L∞(G) comes equipped with an α-doublingintrinsic Markov metric Q determined by an increasing family of projections pj,t asabove. Then, a transferred map T (with amplification for which (4.8) holds) will bebounded from AM to BMOc

Sπprovided :

i) T : Lc2(G)⊗M → Lc

2(G)⊗M is bounded,

ii)(ψ ⊗ ψ ⊗ idM)

ψ(pj,t)2

((pj,t ⊗ pj,t ⊗ 1M)

∣∣δG(T (π(f)p⊥α(j),t))∣∣2) ≲ ∥f∥2M.

Proof. We use the algebraic skeleton(M,Nρ = Nπ, ρ1, ρ2,Eρ, qj,t

)=(M, L∞(G)⊗M,1⊗ ·, π, ψ ⊗ idG, pj,t ⊗ 1G

).

Identity (4.8) is the compatibility condition (2.3). Let us justify the algebraicconditions. The second one is trivial since both Eρ(qj,t) and ρ1(γj,t) belong to R+


in this case. For the first one, consider the product • in L∞(G)⊗Mop. Then, wejust observe that

Eρ(qj,t|ξ|2qj,t) = (ψ ⊗ idM)((pj,t ⊗ 1)ξ∗ξ

)= (ψ ⊗ idM)

(ξ • (pj,t ⊗ 1) • ξ∗

)≤ (ψ ⊗ idM)(ξ • ξ∗) = (ψ ⊗ idM)(ξ∗ξ) = Eρ(|ξ|2).

Define the amplifications

Rπ,j,t : L∞(G)⊗M ∋ ξ 7→ 1

ψ(pj,t)(ψ ⊗ idM)

((pj,t ⊗ 1M)ξ

)∈ M.

Consider also the cpu maps

Φj,t : L∞(G)⊗L∞(G)⊗M → M,

Φj,t(η) =1

ψ(pj,t)2(ψ ⊗ ψ ⊗ idM)

((pj,t ⊗ pj,t ⊗ 1M)η

).

Recalling that δG(x) = x⊗ 1− 1⊗ x, the identity

Φj,t(|δG(ξ)|2) = 2Rπ,j,t(|ξ|2)− 2∣∣Rπ,j,t(ξ)

∣∣2.is straightforward. This readily implies the first analytic condition. On the otherhand, since the auxiliary Markov metric is α-doubling, the second analytic conditionreduces to note that

(ψ ⊗ idM)(qα(j),t ⊗ 1M) ≤ cα(ψ ⊗ idM)(qj,t ⊗ 1M).

Thus, according to inequalitty (4.7), the assertion follows from Theorem 2.1.

Remark 4.14. As noticed, the main particular case of Theorem 4.13 arises for(M, π) = (G,∆) with amplification T ⊗ idG. Condition (4.8) becomes the identity

∆ T = (T ⊗ idG) ∆.

In other words, these are translation invariant CZ operators. We also call themquantum Fourier multipliers in this paper and it can be checked, as expected, thatthese maps are of convolution type in the sense that there exists a kernel k affiliatedto L∞(G) so that

Tf = k ⋆ f = (idG ⊗ ψ)(∆(k)(1G ⊗ Sf)

).

In this particular case, it is not difficult to prove that our conditions in Theorem4.13 reduce to those in Theorem B2 from the Introduction. Of course, Theorem4.13 also applies as well for nonconvolution CZ operators on quantum groups, oreven for transferred forms of them to other von Neumann algebras M.

Remark 4.15. One may consider twisted convolution CZO’s on quantum groupsapplying Theorem 4.13. As an illustration, assume that G L∞(G) by a tracepreserving action α and that G is a quantum group satisfying (αg⊗αg)∆ = ∆αg forall g ∈ G. This property is quite natural in the commutative case, where quantumgroups come from locally compact groups and α is typically implemented by ameasure preserving map β. Note that the underlying Haar measure is translationinvariant and the condition above just imposes that β is an homomorphism. Letus see what we get for a map∑

gfgλ(g) 7→

∑gTg(fg)λ(g),


where the Tg’s are normal convolution maps on L∞(G). Assume L∞(G) comesequipped with a convolution G-equivariant semigroup S∆ which admits a η-doublingintrinsic Markov metric. Then, we get a bounded map L∞(G)⋊.G → BMOc

S⋊when

the following conditions hold:

i) We have a bounded map

Lc2(G)⊗B(ℓ2(G)) ∋ ξ 7→

(Tgh−1

)• ξ ∈ Lc

2(G)⊗B(ℓ2(G)),

where • stands once more for the generalized Schur product of matrices.

ii) Letting R = L∞(G)⊗B(ℓ2(G)) and Ψ(ξ) =∑

g,h(αg−1) • (Tgh−1) • (αg) • ξ,

(ψ ⊗ ψ ⊗ idB(ℓ2(G)))

ψ(pj,t)2

((pj,t ⊗ pj,t ⊗ 1)

∣∣δG(Ψ(ξq⊥η(j),t)∣∣2) ≲ ∥ξ∥2R.

Remark 4.16. All our results in this paragraph impose the additional assumptionthat our quantum groups admit a tracial Haar weight. We believe however that ourresults can be extended to the general non-tracial case. We leave this generalizationopen to the interested reader.

5. Noncommutative transference

Originally motivated by Cotlar’s paper [15] and the method of rotations, Calderondeveloped a circle of ideas [5] which was called the transference method after thesystematic study of Coifman/Weiss in their monograph [13]. The fundamental workof K. de Leeuw [18] also had a big impact in this line of research. Let us consideran amenable locally compact group G with left Haar measure µ, a σ-finite measurespace (Ω, ν) and a uniformly bounded representation β : G → B(Lp(Ω)). Roughly,Calderon’s transference is a technique which allows to transfer the Lp boundednessof a convolution operator f 7→ k ⋆ f on Lp(G) to the corresponding transferredoperator on Lp(Ω)

Vf(w) =

∫G

k(g)βg−1f(w) dµ(g),

for some compactly supported kernel k in L1(G). A case by case limiting procedurealso allows to consider more general (singular) kernels. In the rest of this sectionwe shall develop a noncommutative form of Calderon-Coifman-Weiss technique.

Our first task is to clarify what we mean by ‘representation’ and ‘amenable’ in thecontext of quantum groups. Using the commutative locally compact quantum groupL∞(G) as above, a representation β : G → Aut(M) induces a ∗-representationπβ : M → L∞(G;M) by

πβf(g) = βg−1f.

Note that we have

(idG ⊗ πβ)(πβf)(g, h) = πβ(βg−1f)(h) = βh−1βg−1f

= β(gh)−1f = (∆G ⊗ idM)(πβf)(g, h).

Given a semifinite von Neumann algebra (M, τ) and a locally compact quantumgroup G, this leads us to consider corepresentations π : M 7→ L∞(G)⊗M satisfying(idG ⊗ π) π = (∆⊗ idM) π. Note that comultiplication is a corepresentation bycoassociativity. To show what we mean by ‘uniformly bounded’, let us go back to


our motivating example β : G → Aut(M), where we take M = L∞(Ω) for someσ-finite measure space (Ω, ν). In the classical case

∥βgf∥p ∼ ∥f∥p for all g ∈ G

up to an absolute constant independent of f, g. We say that a corepresentationπ : M → L∞(G)⊗M is uniformly bounded in Lp(M) if for any f ∈ M ∩ Lp(M)we have

1

cπ∥f∥pLp(M) ≤ (idG ⊗ τ)

(|π(f)|p

)≤ cπ∥f∥pLp(M)

for some absolute constant cπ independent of f . Note that our notion again reducesto the classical one on L∞(G). Note also that, since |π(f)|p = π(|f |p), our definitionreduces to the p-independent condition

1

cπ∥f∥L1(M) ≤ (idG ⊗ τ)

(π(f)

)≤ cπ∥f∥L1(M) for all f ∈ M+ ∩ L1(M).

Now we introduce what we mean by an ‘amenable’ quantum group. We say that Gsatisfies Følner’s condition if for every projection q ∈ L1(G) and every ε > 0, thereexists two projections q1, q2 ∈ L1(G) such that

∆(q1)(q ⊗ q2) = q ⊗ q2 and ψ(q1) ≤ (1 + ε)ψ(q2).

In the standard example for a locally compact group G, where (L∞(G), ψ) is L∞(G)equipped with the left Haar measure µ and ∆ is given by ∆G(ξ)(g, h) = ξ(gh) theclassical comultiplication, it turns out that G is amenable iff G is an amenablegroup. Indeed, our notion can be rephrased in this case by saying that for anycompact set K in G and any ε > 0, there exists a neighborhood of the identity Wof finite measure such that

µ(KW) ≤ (1 + ε)µ(W),

which corresponds to (q, q1, q2) = (χK, χKW, χW) in our formulation. This is exactlythe classical characterization of amenability, known as Følner’s condition, used byCoifman and Weiss in [13]. Given an amenable locally compact group G withleft Haar measure µ, it is clear that L∞(G, µ) with its natural quantum groupstructure is amenable. On the other hand, as expected, any compact quantumgroup is amenable just by taking q1 = q2 = 1G.

Assume that G admits a corepresentation π : M → L∞(G)⊗M. Given AM aweakly dense ∗-subalgebra of M, we say that a linear operator V : AM → M is atransferred convolution map if there exists Φ : D ⊂ L∞(G)⊗M → L∞(G)⊗M, anauxiliary convolution map such that π V = Φ π|AM

. The classical transferredoperator

V =

∫G

k(g)βg−1f(w) dµ(g)

comes from

Φ(ξ)(g, w) =

∫G

k(h) ξ(hg,w) dµ(h) = (φ⊗ idG ⊗ idΩ)(∆G ⊗ idΩ).

If πβf(g) = βg−1f denotes the corresponding corepresentation, we may then applythe identities in the proof of Lemma 4.11 again to deduce the following identities

Φπβ = (φ⊗idG⊗idΩ)(∆G⊗idΩ)πβ = (φ⊗idG⊗idΩ)(idG⊗πβ)πβ = πβ(φ⊗idΩ)πβ .


By injectivity of πβ , we must have

Vf(w) = (φ⊗ idΩ)πβf(w) =

∫G

k(g)βg−1f(w) dµ(g)

as expected. This shows how we recover the classical construction.

Let us now settle the framework for our transference result. Assume that G isamenable and consider π : M → L∞(G)⊗M a uniformly bounded corepresentationin some noncommutative measure space (M, τ). We say that T : Lp(G) → Lp(G)is a convolution map with finitely supported L1 kernel when the map T has theform T = (ϕ⊗ idG) ∆ for some functional ϕ = ψ(d ·), with d an element in L1(G)whose left support q satisfies ψ(q) <∞. In the commutative case, this is the kind ofoperators which are transferred. Roughly, the goal is to show how a limit operatorT = limγ Tγ of such maps which is bounded on L2(G) and L∞(G) → BMOS canbe transferred under suitable conditions to a bounded map on Lp(M).

Remark 5.1. Young’s inequality extends to this setting as

∥d ⋆ f∥p ‘=’ ∥(ϕ⊗ idG)∆(f)∥p ≤ 4 ∥d∥1∥f∥p,where ϕ = ψ(d ·) and 1 ≤ p ≤ ∞. Indeed, when d and f are positive the inequalityholds with constant 1. This can by justified by interpolation. When p = 1 we useFubini and the left-invariance of ψ, while for p = ∞ it follows from the fact that(ϕ⊗ idG)∆ is a positive map with 1G 7→ ψ(d). In the general case, we split d, f intotheir positive parts and obtain the constant 4. In fact, the same argument still holdsafter matrix amplification and we deduce that (ϕ ⊗ idG)∆ is completely boundedon Lp(G) with cb-norm 4∥d∥1. This is however not enough for transference, sincethe norms ∥dγ∥1 might not be uniformly bounded.

Theorem 5.2. Let G be an amenable quantum group and consider a uniformlybounded corepresentation π : M → L∞(G)⊗M in some noncommutative measurespace (M, τ). Let T : L2(G) → L2(G) be a bounded map and assume that

(T ⊗ idM) = SOT− limγ(Tγ ⊗ idM)

for some net Tγ = (ϕγ ⊗ idG) ∆ of convolution maps with finitely supported L1

kernels and such that limγ ∥Tγ∥B(L2(G)) ≤ ∥T∥B(L2(G)). Then, the net of transferredoperators Vγ = (ϕγ ⊗ idM) π satisfies the inequalities

∥Vγ∥B(L2(M)) ≤ cπ∥Tγ∥B(L2(G)).

We thus find a WOT-cluster point V satisfying ∥V ∥B(L2(M)) ≤ cπ∥T∥B(L2(G)).

Proof. Note that we have

πVγ = (ϕγ ⊗ id)(idG ⊗ π)π = (ϕγ ⊗ id)(∆⊗ idM)π = (Tγ ⊗ idM)π.

Hence, the uniform boundedness of π yields

1

cπ∥Vγf∥22 ≤ (ρ⊗ τ)

(|πVγ(f)|2

)= (ρ⊗ τ)

(|(Tγ ⊗ idM)π(f)|2

)for any state ρ on L∞(G). On the other hand, if ϕγ = ψ(dγ ·) and qγ denotes theleft support of dγ , we know from the amenability assumption that for any ε > 0 wemay find projections q1γ and q2γ such that

∆(q1γ)(qγ ⊗ q2γ) = qγ ⊗ q2γ and ψ(q1γ) ≤ (1 + ε)ψ(q2γ).


Taking ρ = ψ(q2γ ·)/ψ(q2γ), we obtain the inequality

1

cπ∥Vγf∥22 ≤ (ψ ⊗ τ)

ψ(q2γ)

(∣∣∣(ϕ⊗ id)((∆⊗ idM)π(f)(1G ⊗ q2γ ⊗ 1M)

)∣∣∣2)since ρ is supported by q2γ . Moreover, dγ ⊗ q2γ is supported on the left by qγ ⊗ q2γand amenability provides dγ ⊗ q2γ = ∆(q1γ)(dγ ⊗ q2γ). Once we have created∆(q1γ), we can eliminate q2γ . Altogether gives

1

cπ∥Vγf∥22 ≤ 1

ψ(q2γ)

∥∥∥(Tγ ⊗ id)(π(f)(q1γ ⊗ 1M)

)∥∥∥2L2(L∞(G)⊗M)

.

Now we use the L2 boundedness of Tγ and uniform boundedness of π to conclude

1

cπ∥Vγf∥22 ≤ 1

ψ(q2γ)∥Tγ∥2B(L2(G)) ψ

((q1γ ⊗ 1M)(idG ⊗ τ)

(|π(f)|2

))≤ cπ

ψ(q2γ)∥Tγ∥2B(L2(G)) ψ(q1γ) ∥f∥

22 ≤ cπ (1 + ε) ∥Tγ∥2B(L2(G)) ∥f∥

22.

Letting ε→ 0, we prove the inequality

∥Vγ∥B(L2(M)) ≤ cπ ∥Tγ∥B(L2(G)).

Since T is bounded on L2(G) and limγ ∥Tγ∥B(L2(G)) ≤ ∥T∥B(L2(G)), the operatorsVγ are eventually in a ball of radius cπ(1+δ)∥T∥B(L2(G)) for any δ > 0. The closureof such ball is weak operator compact and thus we find our cluster point.

We now study L∞ → BMO transference and then interpolate/dualize to obtainLp-transference. This approach seems to be new even in the classical theory andwhere our semigroup formulation becomes an essential ingredient.

Corollary 5.3. Let G be a compact (hence amenable) quantum group equippedwith a uniformly bounded corepresentation π : M → L∞(G)⊗M. Let (ϕt)t≥0 bea convolution semigroup of states on L∞(G), giving rise to Markov semigroupsS∆ on (G, ψ) and Sπ on (M, τ). Let T = SOT − limγ Tγ be as above and takeAM = M∩L2(M). Then, if T : L∞(G) → BMOS∆

is completely bounded, we findthat

V = WOT− limγVγ : AM → BMOSπ

is completely bounded. Moreover, if Tπ is regular, the complete boundedness ofJpV : Lp(M) → Lp(M) follows for every 2 < p < ∞ by interpolation. In additionthe complete boundedness of V Jp : Lp(M) → Lp(M) for 1 < p < 2 holds under thesame assumptions for T ∗.

Proof. By uniform boundedness of π we have∥∥∥(idG ⊗ τ)(|π(f)|2

) 12

∥∥∥L∞(G)

≤ cπ∥f∥2,

which implies that π : L2(M) → L∞(G)⊗Lc2(M) is bounded by cπ. According to

the finiteness of L∞(G), we deduce that in fact π : L2(M) → L2(L∞(G)⊗M) isstill bounded with the same norm. This proves that

πV = WOT− limγπVγ = WOT− lim

γ(Tγ ⊗ idM)π = (T ⊗ idM)π.


In particular, πV = (T ⊗ idM)π over AM and identity πSπ,t = (St ⊗ idM)π yields

∥V f∥BMOcSπ

= supt≥0

∥∥∥Sπ,t|V f |2 − |Sπ,tV f |2∥∥∥ 1

2

M

= supt≥0

∥∥∥πSπ,t|V f |2 − |πSπ,tV f |2∥∥∥ 1

2

L∞(G)⊗M

=∥∥(T ⊗ idM)π(f)

∥∥BMOc

S≤ ∥T∥cb∥π(f)∥L∞(G)⊗M = ∥T∥cb∥f∥M

for f ∈ AM. Since the same inequality holds after matrix amplification, we deducethat V : AM → BMOc

Sπis completely bounded with cb-norm ≤ ∥T∥cb. The row

case is similar because

πV † = (πV )† = (Tπ)† = T †π.

The assertions on Lp boundedness follow as usual from Theorem 1.3. Remark 5.4. Under the above assumptions, we see that for V = WOT− limγ Vγwe can find the concrete form of its amplification map Φ defined on L∞(G)⊗M. Inthis case, by applying Theorems 4.13 to Φ = T ⊗ idM, we get Calderon-Zygmundextrapolation for the transferred convolution map V on M.

Acknowledgement. Mei is partially supported by NSF grant DMS1700171.Javier Parcet is supported by the Europa Excelencia Grant MTM2016-81700-ERCand the CSIC Grant PIE-201650E030. Javier Parcet and Runlian Xia are supportedby ICMAT Severo Ochoa Grant SEV-2015-0554 (Spain).

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Marius JungeDepartment of Mathematics

University of Illinois at Urbana-Champaign1409 W. Green St. Urbana, IL 61891. USA

[email protected]

Tao MeiDepartment of Mathematics

Baylor University1301 S University Parks Dr, Waco, TX 76798, USA

Tao [email protected]

Javier ParcetInstituto de Ciencias Matematicas

Consejo Superior de Investigaciones CientıficasC/ Nicolas Cabrera 13-15. 28049, Madrid. Spain

[email protected]

Runlian XiaInstituto de Ciencias Matematicas

Consejo Superior de Investigaciones CientıficasC/ Nicolas Cabrera 13-15. 28049, Madrid. Spain

[email protected]

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